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Theorem axdc3lem2 9233
Description: Lemma for axdc3 9236. We have constructed a "candidate set" 𝑆, which consists of all finite sequences 𝑠 that satisfy our property of interest, namely 𝑠(𝑥 + 1) ∈ 𝐹(𝑠(𝑥)) on its domain, but with the added constraint that 𝑠(0) = 𝐶. These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 9228 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (𝑛):𝑚𝐴 (for some integer 𝑚). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 9228 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that given the sequence , we can construct the sequence 𝑔 that we are after. (Contributed by Mario Carneiro, 30-Jan-2013.)
Hypotheses
Ref Expression
axdc3lem2.1 𝐴 ∈ V
axdc3lem2.2 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
axdc3lem2.3 𝐺 = (𝑥𝑆 ↦ {𝑦𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)})
Assertion
Ref Expression
axdc3lem2 (∃(:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
Distinct variable groups:   𝐴,𝑔,   𝐴,𝑛,𝑠   𝐶,𝑔,   𝐶,𝑛,𝑠   𝑔,𝐹,   𝑛,𝐹,𝑠   𝑘,𝐺   𝑆,𝑘,𝑠   𝑥,𝑆,𝑦   𝑔,𝑘,   ,𝑠   𝑥,,𝑦   𝑘,𝑛
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑘)   𝐶(𝑥,𝑦,𝑘)   𝑆(𝑔,,𝑛)   𝐹(𝑥,𝑦,𝑘)   𝐺(𝑥,𝑦,𝑔,,𝑛,𝑠)

Proof of Theorem axdc3lem2
Dummy variables 𝑖 𝑗 𝑚 𝑢 𝑣 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . . . . . . . . 13 (𝑚 = ∅ → 𝑚 = ∅)
2 fveq2 6158 . . . . . . . . . . . . . 14 (𝑚 = ∅ → (𝑚) = (‘∅))
32dmeqd 5296 . . . . . . . . . . . . 13 (𝑚 = ∅ → dom (𝑚) = dom (‘∅))
41, 3eleq12d 2692 . . . . . . . . . . . 12 (𝑚 = ∅ → (𝑚 ∈ dom (𝑚) ↔ ∅ ∈ dom (‘∅)))
5 eleq2 2687 . . . . . . . . . . . . 13 (𝑚 = ∅ → (𝑗𝑚𝑗 ∈ ∅))
62sseq2d 3618 . . . . . . . . . . . . 13 (𝑚 = ∅ → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (‘∅)))
75, 6imbi12d 334 . . . . . . . . . . . 12 (𝑚 = ∅ → ((𝑗𝑚 → (𝑗) ⊆ (𝑚)) ↔ (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅))))
84, 7anbi12d 746 . . . . . . . . . . 11 (𝑚 = ∅ → ((𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚))) ↔ (∅ ∈ dom (‘∅) ∧ (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅)))))
9 id 22 . . . . . . . . . . . . 13 (𝑚 = 𝑖𝑚 = 𝑖)
10 fveq2 6158 . . . . . . . . . . . . . 14 (𝑚 = 𝑖 → (𝑚) = (𝑖))
1110dmeqd 5296 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → dom (𝑚) = dom (𝑖))
129, 11eleq12d 2692 . . . . . . . . . . . 12 (𝑚 = 𝑖 → (𝑚 ∈ dom (𝑚) ↔ 𝑖 ∈ dom (𝑖)))
13 elequ2 2001 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → (𝑗𝑚𝑗𝑖))
1410sseq2d 3618 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (𝑖)))
1513, 14imbi12d 334 . . . . . . . . . . . 12 (𝑚 = 𝑖 → ((𝑗𝑚 → (𝑗) ⊆ (𝑚)) ↔ (𝑗𝑖 → (𝑗) ⊆ (𝑖))))
1612, 15anbi12d 746 . . . . . . . . . . 11 (𝑚 = 𝑖 → ((𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚))) ↔ (𝑖 ∈ dom (𝑖) ∧ (𝑗𝑖 → (𝑗) ⊆ (𝑖)))))
17 id 22 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖𝑚 = suc 𝑖)
18 fveq2 6158 . . . . . . . . . . . . . 14 (𝑚 = suc 𝑖 → (𝑚) = (‘suc 𝑖))
1918dmeqd 5296 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → dom (𝑚) = dom (‘suc 𝑖))
2017, 19eleq12d 2692 . . . . . . . . . . . 12 (𝑚 = suc 𝑖 → (𝑚 ∈ dom (𝑚) ↔ suc 𝑖 ∈ dom (‘suc 𝑖)))
21 eleq2 2687 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → (𝑗𝑚𝑗 ∈ suc 𝑖))
2218sseq2d 3618 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (‘suc 𝑖)))
2321, 22imbi12d 334 . . . . . . . . . . . 12 (𝑚 = suc 𝑖 → ((𝑗𝑚 → (𝑗) ⊆ (𝑚)) ↔ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖))))
2420, 23anbi12d 746 . . . . . . . . . . 11 (𝑚 = suc 𝑖 → ((𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚))) ↔ (suc 𝑖 ∈ dom (‘suc 𝑖) ∧ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖)))))
25 peano1 7047 . . . . . . . . . . . . . . 15 ∅ ∈ ω
26 ffvelrn 6323 . . . . . . . . . . . . . . 15 ((:ω⟶𝑆 ∧ ∅ ∈ ω) → (‘∅) ∈ 𝑆)
2725, 26mpan2 706 . . . . . . . . . . . . . 14 (:ω⟶𝑆 → (‘∅) ∈ 𝑆)
28 axdc3lem2.2 . . . . . . . . . . . . . . . . . 18 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
29 fdm 6018 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠:suc 𝑛𝐴 → dom 𝑠 = suc 𝑛)
30 nnord 7035 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ω → Ord 𝑛)
31 0elsuc 6997 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Ord 𝑛 → ∅ ∈ suc 𝑛)
3230, 31syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ω → ∅ ∈ suc 𝑛)
33 peano2 7048 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ω → suc 𝑛 ∈ ω)
34 eleq2 2687 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝑠 = suc 𝑛 → (∅ ∈ dom 𝑠 ↔ ∅ ∈ suc 𝑛))
35 eleq1 2686 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝑠 = suc 𝑛 → (dom 𝑠 ∈ ω ↔ suc 𝑛 ∈ ω))
3634, 35anbi12d 746 . . . . . . . . . . . . . . . . . . . . . . . . 25 (dom 𝑠 = suc 𝑛 → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ suc 𝑛 ∧ suc 𝑛 ∈ ω)))
3736biimprcd 240 . . . . . . . . . . . . . . . . . . . . . . . 24 ((∅ ∈ suc 𝑛 ∧ suc 𝑛 ∈ ω) → (dom 𝑠 = suc 𝑛 → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)))
3832, 33, 37syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ω → (dom 𝑠 = suc 𝑛 → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)))
3929, 38syl5com 31 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠:suc 𝑛𝐴 → (𝑛 ∈ ω → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)))
40393ad2ant1 1080 . . . . . . . . . . . . . . . . . . . . 21 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑛 ∈ ω → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)))
4140impcom 446 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))) → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω))
4241rexlimiva 3023 . . . . . . . . . . . . . . . . . . 19 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω))
4342ss2abi 3659 . . . . . . . . . . . . . . . . . 18 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)}
4428, 43eqsstri 3620 . . . . . . . . . . . . . . . . 17 𝑆 ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)}
4544sseli 3584 . . . . . . . . . . . . . . . 16 ((‘∅) ∈ 𝑆 → (‘∅) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)})
46 fvex 6168 . . . . . . . . . . . . . . . . 17 (‘∅) ∈ V
47 dmeq 5294 . . . . . . . . . . . . . . . . . . 19 (𝑠 = (‘∅) → dom 𝑠 = dom (‘∅))
4847eleq2d 2684 . . . . . . . . . . . . . . . . . 18 (𝑠 = (‘∅) → (∅ ∈ dom 𝑠 ↔ ∅ ∈ dom (‘∅)))
4947eleq1d 2683 . . . . . . . . . . . . . . . . . 18 (𝑠 = (‘∅) → (dom 𝑠 ∈ ω ↔ dom (‘∅) ∈ ω))
5048, 49anbi12d 746 . . . . . . . . . . . . . . . . 17 (𝑠 = (‘∅) → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ dom (‘∅) ∧ dom (‘∅) ∈ ω)))
5146, 50elab 3338 . . . . . . . . . . . . . . . 16 ((‘∅) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} ↔ (∅ ∈ dom (‘∅) ∧ dom (‘∅) ∈ ω))
5245, 51sylib 208 . . . . . . . . . . . . . . 15 ((‘∅) ∈ 𝑆 → (∅ ∈ dom (‘∅) ∧ dom (‘∅) ∈ ω))
5352simpld 475 . . . . . . . . . . . . . 14 ((‘∅) ∈ 𝑆 → ∅ ∈ dom (‘∅))
5427, 53syl 17 . . . . . . . . . . . . 13 (:ω⟶𝑆 → ∅ ∈ dom (‘∅))
55 noel 3901 . . . . . . . . . . . . . 14 ¬ 𝑗 ∈ ∅
5655pm2.21i 116 . . . . . . . . . . . . 13 (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅))
5754, 56jctir 560 . . . . . . . . . . . 12 (:ω⟶𝑆 → (∅ ∈ dom (‘∅) ∧ (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅))))
5857adantr 481 . . . . . . . . . . 11 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (∅ ∈ dom (‘∅) ∧ (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅))))
59 ffvelrn 6323 . . . . . . . . . . . . . . 15 ((:ω⟶𝑆𝑖 ∈ ω) → (𝑖) ∈ 𝑆)
6059ancoms 469 . . . . . . . . . . . . . 14 ((𝑖 ∈ ω ∧ :ω⟶𝑆) → (𝑖) ∈ 𝑆)
6160adantrr 752 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (𝑖) ∈ 𝑆)
62 suceq 5759 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑖 → suc 𝑘 = suc 𝑖)
6362fveq2d 6162 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑖 → (‘suc 𝑘) = (‘suc 𝑖))
64 fveq2 6158 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑖 → (𝑘) = (𝑖))
6564fveq2d 6162 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑖 → (𝐺‘(𝑘)) = (𝐺‘(𝑖)))
6663, 65eleq12d 2692 . . . . . . . . . . . . . . 15 (𝑘 = 𝑖 → ((‘suc 𝑘) ∈ (𝐺‘(𝑘)) ↔ (‘suc 𝑖) ∈ (𝐺‘(𝑖))))
6766rspcva 3297 . . . . . . . . . . . . . 14 ((𝑖 ∈ ω ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (‘suc 𝑖) ∈ (𝐺‘(𝑖)))
6867adantrl 751 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (‘suc 𝑖) ∈ (𝐺‘(𝑖)))
6944sseli 3584 . . . . . . . . . . . . . . . . . . . 20 ((𝑖) ∈ 𝑆 → (𝑖) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)})
70 fvex 6168 . . . . . . . . . . . . . . . . . . . . 21 (𝑖) ∈ V
71 dmeq 5294 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 = (𝑖) → dom 𝑠 = dom (𝑖))
7271eleq2d 2684 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = (𝑖) → (∅ ∈ dom 𝑠 ↔ ∅ ∈ dom (𝑖)))
7371eleq1d 2683 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = (𝑖) → (dom 𝑠 ∈ ω ↔ dom (𝑖) ∈ ω))
7472, 73anbi12d 746 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = (𝑖) → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ dom (𝑖) ∧ dom (𝑖) ∈ ω)))
7570, 74elab 3338 . . . . . . . . . . . . . . . . . . . 20 ((𝑖) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} ↔ (∅ ∈ dom (𝑖) ∧ dom (𝑖) ∈ ω))
7669, 75sylib 208 . . . . . . . . . . . . . . . . . . 19 ((𝑖) ∈ 𝑆 → (∅ ∈ dom (𝑖) ∧ dom (𝑖) ∈ ω))
7776simprd 479 . . . . . . . . . . . . . . . . . 18 ((𝑖) ∈ 𝑆 → dom (𝑖) ∈ ω)
78 nnord 7035 . . . . . . . . . . . . . . . . . 18 (dom (𝑖) ∈ ω → Ord dom (𝑖))
79 ordsucelsuc 6984 . . . . . . . . . . . . . . . . . 18 (Ord dom (𝑖) → (𝑖 ∈ dom (𝑖) ↔ suc 𝑖 ∈ suc dom (𝑖)))
8077, 78, 793syl 18 . . . . . . . . . . . . . . . . 17 ((𝑖) ∈ 𝑆 → (𝑖 ∈ dom (𝑖) ↔ suc 𝑖 ∈ suc dom (𝑖)))
8180adantr 481 . . . . . . . . . . . . . . . 16 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (𝑖 ∈ dom (𝑖) ↔ suc 𝑖 ∈ suc dom (𝑖)))
82 dmeq 5294 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = (𝑖) → dom 𝑥 = dom (𝑖))
83 suceq 5759 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝑥 = dom (𝑖) → suc dom 𝑥 = suc dom (𝑖))
8482, 83syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝑖) → suc dom 𝑥 = suc dom (𝑖))
8584eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑖) → (dom 𝑦 = suc dom 𝑥 ↔ dom 𝑦 = suc dom (𝑖)))
8682reseq2d 5366 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝑖) → (𝑦 ↾ dom 𝑥) = (𝑦 ↾ dom (𝑖)))
87 id 22 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝑖) → 𝑥 = (𝑖))
8886, 87eqeq12d 2636 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑖) → ((𝑦 ↾ dom 𝑥) = 𝑥 ↔ (𝑦 ↾ dom (𝑖)) = (𝑖)))
8985, 88anbi12d 746 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑖) → ((dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥) ↔ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))))
9089rabbidv 3181 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (𝑖) → {𝑦𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)} = {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))})
91 axdc3lem2.3 . . . . . . . . . . . . . . . . . . . . . 22 𝐺 = (𝑥𝑆 ↦ {𝑦𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)})
92 axdc3lem2.1 . . . . . . . . . . . . . . . . . . . . . . . 24 𝐴 ∈ V
9392, 28axdc3lem 9232 . . . . . . . . . . . . . . . . . . . . . . 23 𝑆 ∈ V
9493rabex 4783 . . . . . . . . . . . . . . . . . . . . . 22 {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))} ∈ V
9590, 91, 94fvmpt 6249 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖) ∈ 𝑆 → (𝐺‘(𝑖)) = {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))})
9695eleq2d 2684 . . . . . . . . . . . . . . . . . . . 20 ((𝑖) ∈ 𝑆 → ((‘suc 𝑖) ∈ (𝐺‘(𝑖)) ↔ (‘suc 𝑖) ∈ {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))}))
97 dmeq 5294 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (‘suc 𝑖) → dom 𝑦 = dom (‘suc 𝑖))
9897eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (‘suc 𝑖) → (dom 𝑦 = suc dom (𝑖) ↔ dom (‘suc 𝑖) = suc dom (𝑖)))
99 reseq1 5360 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (‘suc 𝑖) → (𝑦 ↾ dom (𝑖)) = ((‘suc 𝑖) ↾ dom (𝑖)))
10099eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (‘suc 𝑖) → ((𝑦 ↾ dom (𝑖)) = (𝑖) ↔ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖)))
10198, 100anbi12d 746 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (‘suc 𝑖) → ((dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖)) ↔ (dom (‘suc 𝑖) = suc dom (𝑖) ∧ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖))))
102101elrab 3351 . . . . . . . . . . . . . . . . . . . 20 ((‘suc 𝑖) ∈ {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))} ↔ ((‘suc 𝑖) ∈ 𝑆 ∧ (dom (‘suc 𝑖) = suc dom (𝑖) ∧ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖))))
10396, 102syl6bb 276 . . . . . . . . . . . . . . . . . . 19 ((𝑖) ∈ 𝑆 → ((‘suc 𝑖) ∈ (𝐺‘(𝑖)) ↔ ((‘suc 𝑖) ∈ 𝑆 ∧ (dom (‘suc 𝑖) = suc dom (𝑖) ∧ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖)))))
104103simplbda 653 . . . . . . . . . . . . . . . . . 18 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (dom (‘suc 𝑖) = suc dom (𝑖) ∧ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖)))
105104simpld 475 . . . . . . . . . . . . . . . . 17 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → dom (‘suc 𝑖) = suc dom (𝑖))
106105eleq2d 2684 . . . . . . . . . . . . . . . 16 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (suc 𝑖 ∈ dom (‘suc 𝑖) ↔ suc 𝑖 ∈ suc dom (𝑖)))
10781, 106bitr4d 271 . . . . . . . . . . . . . . 15 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (𝑖 ∈ dom (𝑖) ↔ suc 𝑖 ∈ dom (‘suc 𝑖)))
108107biimpd 219 . . . . . . . . . . . . . 14 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (𝑖 ∈ dom (𝑖) → suc 𝑖 ∈ dom (‘suc 𝑖)))
109104simprd 479 . . . . . . . . . . . . . . 15 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖))
110 resss 5391 . . . . . . . . . . . . . . . 16 ((‘suc 𝑖) ↾ dom (𝑖)) ⊆ (‘suc 𝑖)
111 sseq1 3611 . . . . . . . . . . . . . . . 16 (((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖) → (((‘suc 𝑖) ↾ dom (𝑖)) ⊆ (‘suc 𝑖) ↔ (𝑖) ⊆ (‘suc 𝑖)))
112110, 111mpbii 223 . . . . . . . . . . . . . . 15 (((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖) → (𝑖) ⊆ (‘suc 𝑖))
113 elsuci 5760 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ suc 𝑖 → (𝑗𝑖𝑗 = 𝑖))
114 pm2.27 42 . . . . . . . . . . . . . . . . . . 19 (𝑗𝑖 → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → (𝑗) ⊆ (𝑖)))
115 sstr2 3595 . . . . . . . . . . . . . . . . . . 19 ((𝑗) ⊆ (𝑖) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖)))
116114, 115syl6 35 . . . . . . . . . . . . . . . . . 18 (𝑗𝑖 → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖))))
117 fveq2 6158 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑖 → (𝑗) = (𝑖))
118117sseq1d 3617 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑖 → ((𝑗) ⊆ (‘suc 𝑖) ↔ (𝑖) ⊆ (‘suc 𝑖)))
119118biimprd 238 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖)))
120119a1d 25 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑖 → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖))))
121116, 120jaoi 394 . . . . . . . . . . . . . . . . 17 ((𝑗𝑖𝑗 = 𝑖) → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖))))
122113, 121syl 17 . . . . . . . . . . . . . . . 16 (𝑗 ∈ suc 𝑖 → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖))))
123122com13 88 . . . . . . . . . . . . . . 15 ((𝑖) ⊆ (‘suc 𝑖) → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖))))
124109, 112, 1233syl 18 . . . . . . . . . . . . . 14 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖))))
125108, 124anim12d 585 . . . . . . . . . . . . 13 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → ((𝑖 ∈ dom (𝑖) ∧ (𝑗𝑖 → (𝑗) ⊆ (𝑖))) → (suc 𝑖 ∈ dom (‘suc 𝑖) ∧ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖)))))
12661, 68, 125syl2anc 692 . . . . . . . . . . . 12 ((𝑖 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → ((𝑖 ∈ dom (𝑖) ∧ (𝑗𝑖 → (𝑗) ⊆ (𝑖))) → (suc 𝑖 ∈ dom (‘suc 𝑖) ∧ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖)))))
127126ex 450 . . . . . . . . . . 11 (𝑖 ∈ ω → ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ((𝑖 ∈ dom (𝑖) ∧ (𝑗𝑖 → (𝑗) ⊆ (𝑖))) → (suc 𝑖 ∈ dom (‘suc 𝑖) ∧ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖))))))
1288, 16, 24, 58, 127finds2 7056 . . . . . . . . . 10 (𝑚 ∈ ω → ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚)))))
129128imp 445 . . . . . . . . 9 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚))))
130129simprd 479 . . . . . . . 8 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (𝑗𝑚 → (𝑗) ⊆ (𝑚)))
131130expcom 451 . . . . . . 7 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ ω → (𝑗𝑚 → (𝑗) ⊆ (𝑚))))
132131ralrimdv 2964 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ ω → ∀𝑗𝑚 (𝑗) ⊆ (𝑚)))
133132ralrimiv 2961 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚))
134 frn 6020 . . . . . . . . . . . 12 (:ω⟶𝑆 → ran 𝑆)
135 ffun 6015 . . . . . . . . . . . . . . . 16 (𝑠:suc 𝑛𝐴 → Fun 𝑠)
1361353ad2ant1 1080 . . . . . . . . . . . . . . 15 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → Fun 𝑠)
137136rexlimivw 3024 . . . . . . . . . . . . . 14 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → Fun 𝑠)
138137ss2abi 3659 . . . . . . . . . . . . 13 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ Fun 𝑠}
13928, 138eqsstri 3620 . . . . . . . . . . . 12 𝑆 ⊆ {𝑠 ∣ Fun 𝑠}
140134, 139syl6ss 3600 . . . . . . . . . . 11 (:ω⟶𝑆 → ran ⊆ {𝑠 ∣ Fun 𝑠})
141140sseld 3587 . . . . . . . . . 10 (:ω⟶𝑆 → (𝑢 ∈ ran 𝑢 ∈ {𝑠 ∣ Fun 𝑠}))
142 vex 3193 . . . . . . . . . . 11 𝑢 ∈ V
143 funeq 5877 . . . . . . . . . . 11 (𝑠 = 𝑢 → (Fun 𝑠 ↔ Fun 𝑢))
144142, 143elab 3338 . . . . . . . . . 10 (𝑢 ∈ {𝑠 ∣ Fun 𝑠} ↔ Fun 𝑢)
145141, 144syl6ib 241 . . . . . . . . 9 (:ω⟶𝑆 → (𝑢 ∈ ran → Fun 𝑢))
146145adantr 481 . . . . . . . 8 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → Fun 𝑢))
147 ffn 6012 . . . . . . . . 9 (:ω⟶𝑆 Fn ω)
148 fvelrnb 6210 . . . . . . . . . . . . 13 ( Fn ω → (𝑣 ∈ ran ↔ ∃𝑏 ∈ ω (𝑏) = 𝑣))
149 fvelrnb 6210 . . . . . . . . . . . . . . 15 ( Fn ω → (𝑢 ∈ ran ↔ ∃𝑎 ∈ ω (𝑎) = 𝑢))
150 nnord 7035 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 ∈ ω → Ord 𝑎)
151 nnord 7035 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 ∈ ω → Ord 𝑏)
152150, 151anim12i 589 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (Ord 𝑎 ∧ Ord 𝑏))
153 ordtri3or 5724 . . . . . . . . . . . . . . . . . . . . . . 23 ((Ord 𝑎 ∧ Ord 𝑏) → (𝑎𝑏𝑎 = 𝑏𝑏𝑎))
154 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 = 𝑏 → (𝑚) = (𝑏))
155154sseq2d 3618 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 = 𝑏 → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (𝑏)))
156155raleqbi1dv 3139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 = 𝑏 → (∀𝑗𝑚 (𝑗) ⊆ (𝑚) ↔ ∀𝑗𝑏 (𝑗) ⊆ (𝑏)))
157156rspcv 3295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 ∈ ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ∀𝑗𝑏 (𝑗) ⊆ (𝑏)))
158 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 = 𝑎 → (𝑗) = (𝑎))
159158sseq1d 3617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = 𝑎 → ((𝑗) ⊆ (𝑏) ↔ (𝑎) ⊆ (𝑏)))
160159rspccv 3296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∀𝑗𝑏 (𝑗) ⊆ (𝑏) → (𝑎𝑏 → (𝑎) ⊆ (𝑏)))
161157, 160syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 ∈ ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑎𝑏 → (𝑎) ⊆ (𝑏))))
162161adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑎𝑏 → (𝑎) ⊆ (𝑏))))
1631623imp 1254 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) ∧ 𝑎𝑏) → (𝑎) ⊆ (𝑏))
164163orcd 407 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) ∧ 𝑎𝑏) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))
1651643exp 1261 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑎𝑏 → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
166165com3r 87 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎𝑏 → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
167 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = 𝑏 → (𝑎) = (𝑏))
168 eqimss 3642 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎) = (𝑏) → (𝑎) ⊆ (𝑏))
169168orcd 407 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎) = (𝑏) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))
170167, 169syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑏 → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))
1711702a1d 26 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑏 → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
172 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 = 𝑎 → (𝑚) = (𝑎))
173172sseq2d 3618 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 = 𝑎 → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (𝑎)))
174173raleqbi1dv 3139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 = 𝑎 → (∀𝑗𝑚 (𝑗) ⊆ (𝑚) ↔ ∀𝑗𝑎 (𝑗) ⊆ (𝑎)))
175174rspcv 3295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎 ∈ ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ∀𝑗𝑎 (𝑗) ⊆ (𝑎)))
176 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 = 𝑏 → (𝑗) = (𝑏))
177176sseq1d 3617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = 𝑏 → ((𝑗) ⊆ (𝑎) ↔ (𝑏) ⊆ (𝑎)))
178177rspccv 3296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∀𝑗𝑎 (𝑗) ⊆ (𝑎) → (𝑏𝑎 → (𝑏) ⊆ (𝑎)))
179175, 178syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 ∈ ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑏𝑎 → (𝑏) ⊆ (𝑎))))
180179adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑏𝑎 → (𝑏) ⊆ (𝑎))))
1811803imp 1254 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) ∧ 𝑏𝑎) → (𝑏) ⊆ (𝑎))
182181olcd 408 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) ∧ 𝑏𝑎) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))
1831823exp 1261 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑏𝑎 → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
184183com3r 87 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏𝑎 → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
185166, 171, 1843jaoi 1388 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎𝑏𝑎 = 𝑏𝑏𝑎) → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
186153, 185syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((Ord 𝑎 ∧ Ord 𝑏) → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
187152, 186mpcom 38 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎))))
188 sseq12 3613 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → ((𝑎) ⊆ (𝑏) ↔ 𝑢𝑣))
189 sseq12 3613 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑏) = 𝑣 ∧ (𝑎) = 𝑢) → ((𝑏) ⊆ (𝑎) ↔ 𝑣𝑢))
190189ancoms 469 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → ((𝑏) ⊆ (𝑎) ↔ 𝑣𝑢))
191188, 190orbi12d 745 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → (((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)) ↔ (𝑢𝑣𝑣𝑢)))
192191biimpcd 239 . . . . . . . . . . . . . . . . . . . . 21 (((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)) → (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → (𝑢𝑣𝑣𝑢)))
193187, 192syl6 35 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → (𝑢𝑣𝑣𝑢))))
194193com23 86 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢))))
195194exp4b 631 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ ω → (𝑏 ∈ ω → ((𝑎) = 𝑢 → ((𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢))))))
196195com23 86 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ ω → ((𝑎) = 𝑢 → (𝑏 ∈ ω → ((𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢))))))
197196rexlimiv 3022 . . . . . . . . . . . . . . . 16 (∃𝑎 ∈ ω (𝑎) = 𝑢 → (𝑏 ∈ ω → ((𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢)))))
198197rexlimdv 3025 . . . . . . . . . . . . . . 15 (∃𝑎 ∈ ω (𝑎) = 𝑢 → (∃𝑏 ∈ ω (𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢))))
199149, 198syl6bi 243 . . . . . . . . . . . . . 14 ( Fn ω → (𝑢 ∈ ran → (∃𝑏 ∈ ω (𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢)))))
200199com23 86 . . . . . . . . . . . . 13 ( Fn ω → (∃𝑏 ∈ ω (𝑏) = 𝑣 → (𝑢 ∈ ran → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢)))))
201148, 200sylbid 230 . . . . . . . . . . . 12 ( Fn ω → (𝑣 ∈ ran → (𝑢 ∈ ran → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢)))))
202201com24 95 . . . . . . . . . . 11 ( Fn ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢 ∈ ran → (𝑣 ∈ ran → (𝑢𝑣𝑣𝑢)))))
203202imp 445 . . . . . . . . . 10 (( Fn ω ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → (𝑣 ∈ ran → (𝑢𝑣𝑣𝑢))))
204203ralrimdv 2964 . . . . . . . . 9 (( Fn ω ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢)))
205147, 204sylan 488 . . . . . . . 8 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢)))
206146, 205jcad 555 . . . . . . 7 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → (Fun 𝑢 ∧ ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢))))
207206ralrimiv 2961 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → ∀𝑢 ∈ ran (Fun 𝑢 ∧ ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢)))
208 fununi 5932 . . . . . 6 (∀𝑢 ∈ ran (Fun 𝑢 ∧ ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢)) → Fun ran )
209207, 208syl 17 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → Fun ran )
210133, 209syldan 487 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → Fun ran )
211 vex 3193 . . . . . . . . 9 𝑚 ∈ V
212211eldm2 5292 . . . . . . . 8 (𝑚 ∈ dom ran ↔ ∃𝑢𝑚, 𝑢⟩ ∈ ran )
213 eluni2 4413 . . . . . . . . . 10 (⟨𝑚, 𝑢⟩ ∈ ran ↔ ∃𝑣 ∈ ran 𝑚, 𝑢⟩ ∈ 𝑣)
214211, 142opeldm 5298 . . . . . . . . . . . . . . 15 (⟨𝑚, 𝑢⟩ ∈ 𝑣𝑚 ∈ dom 𝑣)
215214a1i 11 . . . . . . . . . . . . . 14 (:ω⟶𝑆 → (⟨𝑚, 𝑢⟩ ∈ 𝑣𝑚 ∈ dom 𝑣))
216134, 44syl6ss 3600 . . . . . . . . . . . . . . 15 (:ω⟶𝑆 → ran ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)})
217 ssel 3582 . . . . . . . . . . . . . . . 16 (ran ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} → (𝑣 ∈ ran 𝑣 ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)}))
218 vex 3193 . . . . . . . . . . . . . . . . . 18 𝑣 ∈ V
219 dmeq 5294 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑣 → dom 𝑠 = dom 𝑣)
220219eleq2d 2684 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑣 → (∅ ∈ dom 𝑠 ↔ ∅ ∈ dom 𝑣))
221219eleq1d 2683 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑣 → (dom 𝑠 ∈ ω ↔ dom 𝑣 ∈ ω))
222220, 221anbi12d 746 . . . . . . . . . . . . . . . . . 18 (𝑠 = 𝑣 → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ dom 𝑣 ∧ dom 𝑣 ∈ ω)))
223218, 222elab 3338 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} ↔ (∅ ∈ dom 𝑣 ∧ dom 𝑣 ∈ ω))
224223simprbi 480 . . . . . . . . . . . . . . . 16 (𝑣 ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} → dom 𝑣 ∈ ω)
225217, 224syl6 35 . . . . . . . . . . . . . . 15 (ran ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} → (𝑣 ∈ ran → dom 𝑣 ∈ ω))
226216, 225syl 17 . . . . . . . . . . . . . 14 (:ω⟶𝑆 → (𝑣 ∈ ran → dom 𝑣 ∈ ω))
227215, 226anim12d 585 . . . . . . . . . . . . 13 (:ω⟶𝑆 → ((⟨𝑚, 𝑢⟩ ∈ 𝑣𝑣 ∈ ran ) → (𝑚 ∈ dom 𝑣 ∧ dom 𝑣 ∈ ω)))
228 elnn 7037 . . . . . . . . . . . . 13 ((𝑚 ∈ dom 𝑣 ∧ dom 𝑣 ∈ ω) → 𝑚 ∈ ω)
229227, 228syl6 35 . . . . . . . . . . . 12 (:ω⟶𝑆 → ((⟨𝑚, 𝑢⟩ ∈ 𝑣𝑣 ∈ ran ) → 𝑚 ∈ ω))
230229expcomd 454 . . . . . . . . . . 11 (:ω⟶𝑆 → (𝑣 ∈ ran → (⟨𝑚, 𝑢⟩ ∈ 𝑣𝑚 ∈ ω)))
231230rexlimdv 3025 . . . . . . . . . 10 (:ω⟶𝑆 → (∃𝑣 ∈ ran 𝑚, 𝑢⟩ ∈ 𝑣𝑚 ∈ ω))
232213, 231syl5bi 232 . . . . . . . . 9 (:ω⟶𝑆 → (⟨𝑚, 𝑢⟩ ∈ ran 𝑚 ∈ ω))
233232exlimdv 1858 . . . . . . . 8 (:ω⟶𝑆 → (∃𝑢𝑚, 𝑢⟩ ∈ ran 𝑚 ∈ ω))
234212, 233syl5bi 232 . . . . . . 7 (:ω⟶𝑆 → (𝑚 ∈ dom ran 𝑚 ∈ ω))
235234adantr 481 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ dom ran 𝑚 ∈ ω))
236 id 22 . . . . . . . . . . 11 (𝑚 ∈ ω → 𝑚 ∈ ω)
237 fnfvelrn 6322 . . . . . . . . . . 11 (( Fn ω ∧ 𝑚 ∈ ω) → (𝑚) ∈ ran )
238147, 236, 237syl2anr 495 . . . . . . . . . 10 ((𝑚 ∈ ω ∧ :ω⟶𝑆) → (𝑚) ∈ ran )
239238adantrr 752 . . . . . . . . 9 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (𝑚) ∈ ran )
240129simpld 475 . . . . . . . . 9 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → 𝑚 ∈ dom (𝑚))
241 dmeq 5294 . . . . . . . . . 10 (𝑢 = (𝑚) → dom 𝑢 = dom (𝑚))
242241eliuni 4499 . . . . . . . . 9 (((𝑚) ∈ ran 𝑚 ∈ dom (𝑚)) → 𝑚 𝑢 ∈ ran dom 𝑢)
243239, 240, 242syl2anc 692 . . . . . . . 8 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → 𝑚 𝑢 ∈ ran dom 𝑢)
244 dmuni 5304 . . . . . . . 8 dom ran = 𝑢 ∈ ran dom 𝑢
245243, 244syl6eleqr 2709 . . . . . . 7 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → 𝑚 ∈ dom ran )
246245expcom 451 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ ω → 𝑚 ∈ dom ran ))
247235, 246impbid 202 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ dom ran 𝑚 ∈ ω))
248247eqrdv 2619 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → dom ran = ω)
249 rnuni 5513 . . . . . 6 ran ran = 𝑠 ∈ ran ran 𝑠
250 frn 6020 . . . . . . . . . . . . . 14 (𝑠:suc 𝑛𝐴 → ran 𝑠𝐴)
2512503ad2ant1 1080 . . . . . . . . . . . . 13 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → ran 𝑠𝐴)
252251rexlimivw 3024 . . . . . . . . . . . 12 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → ran 𝑠𝐴)
253252ss2abi 3659 . . . . . . . . . . 11 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ ran 𝑠𝐴}
25428, 253eqsstri 3620 . . . . . . . . . 10 𝑆 ⊆ {𝑠 ∣ ran 𝑠𝐴}
255134, 254syl6ss 3600 . . . . . . . . 9 (:ω⟶𝑆 → ran ⊆ {𝑠 ∣ ran 𝑠𝐴})
256 ssel 3582 . . . . . . . . . 10 (ran ⊆ {𝑠 ∣ ran 𝑠𝐴} → (𝑠 ∈ ran 𝑠 ∈ {𝑠 ∣ ran 𝑠𝐴}))
257 abid 2609 . . . . . . . . . 10 (𝑠 ∈ {𝑠 ∣ ran 𝑠𝐴} ↔ ran 𝑠𝐴)
258256, 257syl6ib 241 . . . . . . . . 9 (ran ⊆ {𝑠 ∣ ran 𝑠𝐴} → (𝑠 ∈ ran → ran 𝑠𝐴))
259255, 258syl 17 . . . . . . . 8 (:ω⟶𝑆 → (𝑠 ∈ ran → ran 𝑠𝐴))
260259ralrimiv 2961 . . . . . . 7 (:ω⟶𝑆 → ∀𝑠 ∈ ran ran 𝑠𝐴)
261 iunss 4534 . . . . . . 7 ( 𝑠 ∈ ran ran 𝑠𝐴 ↔ ∀𝑠 ∈ ran ran 𝑠𝐴)
262260, 261sylibr 224 . . . . . 6 (:ω⟶𝑆 𝑠 ∈ ran ran 𝑠𝐴)
263249, 262syl5eqss 3634 . . . . 5 (:ω⟶𝑆 → ran ran 𝐴)
264263adantr 481 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ran ran 𝐴)
265 df-fn 5860 . . . . 5 ( ran Fn ω ↔ (Fun ran ∧ dom ran = ω))
266 df-f 5861 . . . . . 6 ( ran :ω⟶𝐴 ↔ ( ran Fn ω ∧ ran ran 𝐴))
267266biimpri 218 . . . . 5 (( ran Fn ω ∧ ran ran 𝐴) → ran :ω⟶𝐴)
268265, 267sylanbr 490 . . . 4 (((Fun ran ∧ dom ran = ω) ∧ ran ran 𝐴) → ran :ω⟶𝐴)
269210, 248, 264, 268syl21anc 1322 . . 3 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ran :ω⟶𝐴)
270 fnfvelrn 6322 . . . . . . . 8 (( Fn ω ∧ ∅ ∈ ω) → (‘∅) ∈ ran )
271147, 25, 270sylancl 693 . . . . . . 7 (:ω⟶𝑆 → (‘∅) ∈ ran )
272271adantr 481 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (‘∅) ∈ ran )
273 elssuni 4440 . . . . . 6 ((‘∅) ∈ ran → (‘∅) ⊆ ran )
274272, 273syl 17 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (‘∅) ⊆ ran )
27554adantr 481 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∅ ∈ dom (‘∅))
276 funssfv 6176 . . . . 5 ((Fun ran ∧ (‘∅) ⊆ ran ∧ ∅ ∈ dom (‘∅)) → ( ran ‘∅) = ((‘∅)‘∅))
277210, 274, 275, 276syl3anc 1323 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ( ran ‘∅) = ((‘∅)‘∅))
278 simp2 1060 . . . . . . . . . . 11 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑠‘∅) = 𝐶)
279278rexlimivw 3024 . . . . . . . . . 10 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑠‘∅) = 𝐶)
280279ss2abi 3659 . . . . . . . . 9 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶}
28128, 280eqsstri 3620 . . . . . . . 8 𝑆 ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶}
282134, 281syl6ss 3600 . . . . . . 7 (:ω⟶𝑆 → ran ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶})
283 ssel 3582 . . . . . . . 8 (ran ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶} → ((‘∅) ∈ ran → (‘∅) ∈ {𝑠 ∣ (𝑠‘∅) = 𝐶}))
284 fveq1 6157 . . . . . . . . . 10 (𝑠 = (‘∅) → (𝑠‘∅) = ((‘∅)‘∅))
285284eqeq1d 2623 . . . . . . . . 9 (𝑠 = (‘∅) → ((𝑠‘∅) = 𝐶 ↔ ((‘∅)‘∅) = 𝐶))
28646, 285elab 3338 . . . . . . . 8 ((‘∅) ∈ {𝑠 ∣ (𝑠‘∅) = 𝐶} ↔ ((‘∅)‘∅) = 𝐶)
287283, 286syl6ib 241 . . . . . . 7 (ran ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶} → ((‘∅) ∈ ran → ((‘∅)‘∅) = 𝐶))
288282, 287syl 17 . . . . . 6 (:ω⟶𝑆 → ((‘∅) ∈ ran → ((‘∅)‘∅) = 𝐶))
289288adantr 481 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ((‘∅) ∈ ran → ((‘∅)‘∅) = 𝐶))
290272, 289mpd 15 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ((‘∅)‘∅) = 𝐶)
291277, 290eqtrd 2655 . . 3 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ( ran ‘∅) = 𝐶)
292 nfv 1840 . . . . 5 𝑘 :ω⟶𝑆
293 nfra1 2937 . . . . 5 𝑘𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))
294292, 293nfan 1825 . . . 4 𝑘(:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))
295134ad2antrr 761 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ran 𝑆)
296 peano2 7048 . . . . . . . . 9 (𝑘 ∈ ω → suc 𝑘 ∈ ω)
297 fnfvelrn 6322 . . . . . . . . 9 (( Fn ω ∧ suc 𝑘 ∈ ω) → (‘suc 𝑘) ∈ ran )
298147, 296, 297syl2an 494 . . . . . . . 8 ((:ω⟶𝑆𝑘 ∈ ω) → (‘suc 𝑘) ∈ ran )
299298adantlr 750 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → (‘suc 𝑘) ∈ ran )
300240expcom 451 . . . . . . . . 9 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ ω → 𝑚 ∈ dom (𝑚)))
301300ralrimiv 2961 . . . . . . . 8 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∀𝑚 ∈ ω 𝑚 ∈ dom (𝑚))
302 id 22 . . . . . . . . . . 11 (𝑚 = suc 𝑘𝑚 = suc 𝑘)
303 fveq2 6158 . . . . . . . . . . . 12 (𝑚 = suc 𝑘 → (𝑚) = (‘suc 𝑘))
304303dmeqd 5296 . . . . . . . . . . 11 (𝑚 = suc 𝑘 → dom (𝑚) = dom (‘suc 𝑘))
305302, 304eleq12d 2692 . . . . . . . . . 10 (𝑚 = suc 𝑘 → (𝑚 ∈ dom (𝑚) ↔ suc 𝑘 ∈ dom (‘suc 𝑘)))
306305rspcv 3295 . . . . . . . . 9 (suc 𝑘 ∈ ω → (∀𝑚 ∈ ω 𝑚 ∈ dom (𝑚) → suc 𝑘 ∈ dom (‘suc 𝑘)))
307296, 306syl 17 . . . . . . . 8 (𝑘 ∈ ω → (∀𝑚 ∈ ω 𝑚 ∈ dom (𝑚) → suc 𝑘 ∈ dom (‘suc 𝑘)))
308301, 307mpan9 486 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → suc 𝑘 ∈ dom (‘suc 𝑘))
309 eleq2 2687 . . . . . . . . . . . . . . . . . . . . 21 (dom 𝑠 = suc 𝑛 → (suc 𝑘 ∈ dom 𝑠 ↔ suc 𝑘 ∈ suc 𝑛))
310309biimpa 501 . . . . . . . . . . . . . . . . . . . 20 ((dom 𝑠 = suc 𝑛 ∧ suc 𝑘 ∈ dom 𝑠) → suc 𝑘 ∈ suc 𝑛)
31129, 310sylan 488 . . . . . . . . . . . . . . . . . . 19 ((𝑠:suc 𝑛𝐴 ∧ suc 𝑘 ∈ dom 𝑠) → suc 𝑘 ∈ suc 𝑛)
312 ordsucelsuc 6984 . . . . . . . . . . . . . . . . . . . . . . 23 (Ord 𝑛 → (𝑘𝑛 ↔ suc 𝑘 ∈ suc 𝑛))
31330, 312syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ ω → (𝑘𝑛 ↔ suc 𝑘 ∈ suc 𝑛))
314313biimprd 238 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ω → (suc 𝑘 ∈ suc 𝑛𝑘𝑛))
315 rsp 2925 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑘𝑛 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
316314, 315syl9r 78 . . . . . . . . . . . . . . . . . . . 20 (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑛 ∈ ω → (suc 𝑘 ∈ suc 𝑛 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
317316com13 88 . . . . . . . . . . . . . . . . . . 19 (suc 𝑘 ∈ suc 𝑛 → (𝑛 ∈ ω → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
318311, 317syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑠:suc 𝑛𝐴 ∧ suc 𝑘 ∈ dom 𝑠) → (𝑛 ∈ ω → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
319318ex 450 . . . . . . . . . . . . . . . . 17 (𝑠:suc 𝑛𝐴 → (suc 𝑘 ∈ dom 𝑠 → (𝑛 ∈ ω → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))))
320319com24 95 . . . . . . . . . . . . . . . 16 (𝑠:suc 𝑛𝐴 → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑛 ∈ ω → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))))
321320imp 445 . . . . . . . . . . . . . . 15 ((𝑠:suc 𝑛𝐴 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑛 ∈ ω → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
3223213adant2 1078 . . . . . . . . . . . . . 14 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑛 ∈ ω → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
323322impcom 446 . . . . . . . . . . . . 13 ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))) → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
324323rexlimiva 3023 . . . . . . . . . . . 12 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
325324ss2abi 3659 . . . . . . . . . . 11 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
32628, 325eqsstri 3620 . . . . . . . . . 10 𝑆 ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
327 sstr 3596 . . . . . . . . . 10 ((ran 𝑆𝑆 ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}) → ran ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))})
328326, 327mpan2 706 . . . . . . . . 9 (ran 𝑆 → ran ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))})
329328sseld 3587 . . . . . . . 8 (ran 𝑆 → ((‘suc 𝑘) ∈ ran → (‘suc 𝑘) ∈ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}))
330 fvex 6168 . . . . . . . . 9 (‘suc 𝑘) ∈ V
331 dmeq 5294 . . . . . . . . . . 11 (𝑠 = (‘suc 𝑘) → dom 𝑠 = dom (‘suc 𝑘))
332331eleq2d 2684 . . . . . . . . . 10 (𝑠 = (‘suc 𝑘) → (suc 𝑘 ∈ dom 𝑠 ↔ suc 𝑘 ∈ dom (‘suc 𝑘)))
333 fveq1 6157 . . . . . . . . . . 11 (𝑠 = (‘suc 𝑘) → (𝑠‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘))
334 fveq1 6157 . . . . . . . . . . . 12 (𝑠 = (‘suc 𝑘) → (𝑠𝑘) = ((‘suc 𝑘)‘𝑘))
335334fveq2d 6162 . . . . . . . . . . 11 (𝑠 = (‘suc 𝑘) → (𝐹‘(𝑠𝑘)) = (𝐹‘((‘suc 𝑘)‘𝑘)))
336333, 335eleq12d 2692 . . . . . . . . . 10 (𝑠 = (‘suc 𝑘) → ((𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) ↔ ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘))))
337332, 336imbi12d 334 . . . . . . . . 9 (𝑠 = (‘suc 𝑘) → ((suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) ↔ (suc 𝑘 ∈ dom (‘suc 𝑘) → ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘)))))
338330, 337elab 3338 . . . . . . . 8 ((‘suc 𝑘) ∈ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ↔ (suc 𝑘 ∈ dom (‘suc 𝑘) → ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘))))
339329, 338syl6ib 241 . . . . . . 7 (ran 𝑆 → ((‘suc 𝑘) ∈ ran → (suc 𝑘 ∈ dom (‘suc 𝑘) → ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘)))))
340295, 299, 308, 339syl3c 66 . . . . . 6 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘)))
341210adantr 481 . . . . . . . 8 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → Fun ran )
342 elssuni 4440 . . . . . . . . . 10 ((‘suc 𝑘) ∈ ran → (‘suc 𝑘) ⊆ ran )
343298, 342syl 17 . . . . . . . . 9 ((:ω⟶𝑆𝑘 ∈ ω) → (‘suc 𝑘) ⊆ ran )
344343adantlr 750 . . . . . . . 8 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → (‘suc 𝑘) ⊆ ran )
345 funssfv 6176 . . . . . . . 8 ((Fun ran ∧ (‘suc 𝑘) ⊆ ran ∧ suc 𝑘 ∈ dom (‘suc 𝑘)) → ( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘))
346341, 344, 308, 345syl3anc 1323 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘))
347216sseld 3587 . . . . . . . . . . . . . . 15 (:ω⟶𝑆 → ((‘suc 𝑘) ∈ ran → (‘suc 𝑘) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)}))
348331eleq2d 2684 . . . . . . . . . . . . . . . . 17 (𝑠 = (‘suc 𝑘) → (∅ ∈ dom 𝑠 ↔ ∅ ∈ dom (‘suc 𝑘)))
349331eleq1d 2683 . . . . . . . . . . . . . . . . 17 (𝑠 = (‘suc 𝑘) → (dom 𝑠 ∈ ω ↔ dom (‘suc 𝑘) ∈ ω))
350348, 349anbi12d 746 . . . . . . . . . . . . . . . 16 (𝑠 = (‘suc 𝑘) → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω)))
351330, 350elab 3338 . . . . . . . . . . . . . . 15 ((‘suc 𝑘) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} ↔ (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω))
352347, 351syl6ib 241 . . . . . . . . . . . . . 14 (:ω⟶𝑆 → ((‘suc 𝑘) ∈ ran → (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω)))
353352adantr 481 . . . . . . . . . . . . 13 ((:ω⟶𝑆𝑘 ∈ ω) → ((‘suc 𝑘) ∈ ran → (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω)))
354298, 353mpd 15 . . . . . . . . . . . 12 ((:ω⟶𝑆𝑘 ∈ ω) → (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω))
355354simprd 479 . . . . . . . . . . 11 ((:ω⟶𝑆𝑘 ∈ ω) → dom (‘suc 𝑘) ∈ ω)
356 nnord 7035 . . . . . . . . . . 11 (dom (‘suc 𝑘) ∈ ω → Ord dom (‘suc 𝑘))
357 ordtr 5706 . . . . . . . . . . 11 (Ord dom (‘suc 𝑘) → Tr dom (‘suc 𝑘))
358 trsuc 5779 . . . . . . . . . . . 12 ((Tr dom (‘suc 𝑘) ∧ suc 𝑘 ∈ dom (‘suc 𝑘)) → 𝑘 ∈ dom (‘suc 𝑘))
359358ex 450 . . . . . . . . . . 11 (Tr dom (‘suc 𝑘) → (suc 𝑘 ∈ dom (‘suc 𝑘) → 𝑘 ∈ dom (‘suc 𝑘)))
360355, 356, 357, 3594syl 19 . . . . . . . . . 10 ((:ω⟶𝑆𝑘 ∈ ω) → (suc 𝑘 ∈ dom (‘suc 𝑘) → 𝑘 ∈ dom (‘suc 𝑘)))
361360adantlr 750 . . . . . . . . 9 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → (suc 𝑘 ∈ dom (‘suc 𝑘) → 𝑘 ∈ dom (‘suc 𝑘)))
362308, 361mpd 15 . . . . . . . 8 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → 𝑘 ∈ dom (‘suc 𝑘))
363 funssfv 6176 . . . . . . . 8 ((Fun ran ∧ (‘suc 𝑘) ⊆ ran 𝑘 ∈ dom (‘suc 𝑘)) → ( ran 𝑘) = ((‘suc 𝑘)‘𝑘))
364341, 344, 362, 363syl3anc 1323 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ( ran 𝑘) = ((‘suc 𝑘)‘𝑘))
365 simpl 473 . . . . . . . 8 ((( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘) ∧ ( ran 𝑘) = ((‘suc 𝑘)‘𝑘)) → ( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘))
366 simpr 477 . . . . . . . . 9 ((( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘) ∧ ( ran 𝑘) = ((‘suc 𝑘)‘𝑘)) → ( ran 𝑘) = ((‘suc 𝑘)‘𝑘))
367366fveq2d 6162 . . . . . . . 8 ((( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘) ∧ ( ran 𝑘) = ((‘suc 𝑘)‘𝑘)) → (𝐹‘( ran 𝑘)) = (𝐹‘((‘suc 𝑘)‘𝑘)))
368365, 367eleq12d 2692 . . . . . . 7 ((( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘) ∧ ( ran 𝑘) = ((‘suc 𝑘)‘𝑘)) → (( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)) ↔ ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘))))
369346, 364, 368syl2anc 692 . . . . . 6 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → (( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)) ↔ ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘))))
370340, 369mpbird 247 . . . . 5 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)))
371370ex 450 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑘 ∈ ω → ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘))))
372294, 371ralrimi 2953 . . 3 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∀𝑘 ∈ ω ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)))
373 vex 3193 . . . . . 6 ∈ V
374373rnex 7062 . . . . 5 ran ∈ V
375374uniex 6918 . . . 4 ran ∈ V
376 feq1 5993 . . . . 5 (𝑔 = ran → (𝑔:ω⟶𝐴 ran :ω⟶𝐴))
377 fveq1 6157 . . . . . 6 (𝑔 = ran → (𝑔‘∅) = ( ran ‘∅))
378377eqeq1d 2623 . . . . 5 (𝑔 = ran → ((𝑔‘∅) = 𝐶 ↔ ( ran ‘∅) = 𝐶))
379 fveq1 6157 . . . . . . 7 (𝑔 = ran → (𝑔‘suc 𝑘) = ( ran ‘suc 𝑘))
380 fveq1 6157 . . . . . . . 8 (𝑔 = ran → (𝑔𝑘) = ( ran 𝑘))
381380fveq2d 6162 . . . . . . 7 (𝑔 = ran → (𝐹‘(𝑔𝑘)) = (𝐹‘( ran 𝑘)))
382379, 381eleq12d 2692 . . . . . 6 (𝑔 = ran → ((𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘)) ↔ ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘))))
383382ralbidv 2982 . . . . 5 (𝑔 = ran → (∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘)) ↔ ∀𝑘 ∈ ω ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘))))
384376, 378, 3833anbi123d 1396 . . . 4 (𝑔 = ran → ((𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))) ↔ ( ran :ω⟶𝐴 ∧ ( ran ‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)))))
385375, 384spcev 3290 . . 3 (( ran :ω⟶𝐴 ∧ ( ran ‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
386269, 291, 372, 385syl3anc 1323 . 2 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
387386exlimiv 1855 1 (∃(:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3o 1035  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2908  wrex 2909  {crab 2912  Vcvv 3190  wss 3560  c0 3897  cop 4161   cuni 4409   ciun 4492  cmpt 4683  Tr wtr 4722  dom cdm 5084  ran crn 5085  cres 5086  Ord word 5691  suc csuc 5694  Fun wfun 5851   Fn wfn 5852  wf 5853  cfv 5857  ωcom 7027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-dc 9228
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-om 7028  df-1o 7520
This theorem is referenced by:  axdc3lem4  9235
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