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Theorem fseqenlem2 9038
Description: Lemma for fseqen 9040. (Contributed by Mario Carneiro, 17-May-2015.)
Hypotheses
Ref Expression
fseqenlem.a (𝜑𝐴𝑉)
fseqenlem.b (𝜑𝐵𝐴)
fseqenlem.f (𝜑𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)
fseqenlem.g 𝐺 = seq𝜔((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))), {⟨∅, 𝐵⟩})
fseqenlem.k 𝐾 = (𝑦 𝑘 ∈ ω (𝐴𝑚 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)
Assertion
Ref Expression
fseqenlem2 (𝜑𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)–1-1→(ω × 𝐴))
Distinct variable groups:   𝑦,𝐵   𝑓,𝑛,𝑥,𝐹   𝑦,𝑘,𝐺   𝑓,𝑘,𝑦,𝐴,𝑛,𝑥   𝜑,𝑘,𝑛,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑥,𝑓,𝑘,𝑛)   𝐹(𝑦,𝑘)   𝐺(𝑥,𝑓,𝑛)   𝐾(𝑥,𝑦,𝑓,𝑘,𝑛)   𝑉(𝑥,𝑦,𝑓,𝑘,𝑛)

Proof of Theorem fseqenlem2
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eliun 4676 . . . . 5 (𝑦 𝑘 ∈ ω (𝐴𝑚 𝑘) ↔ ∃𝑘 ∈ ω 𝑦 ∈ (𝐴𝑚 𝑘))
2 elmapi 8045 . . . . . . . . . 10 (𝑦 ∈ (𝐴𝑚 𝑘) → 𝑦:𝑘𝐴)
32ad2antll 767 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → 𝑦:𝑘𝐴)
4 fdm 6212 . . . . . . . . 9 (𝑦:𝑘𝐴 → dom 𝑦 = 𝑘)
53, 4syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → dom 𝑦 = 𝑘)
6 simprl 811 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → 𝑘 ∈ ω)
75, 6eqeltrd 2839 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → dom 𝑦 ∈ ω)
85fveq2d 6356 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → (𝐺‘dom 𝑦) = (𝐺𝑘))
98fveq1d 6354 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺𝑘)‘𝑦))
10 fseqenlem.a . . . . . . . . . . . 12 (𝜑𝐴𝑉)
11 fseqenlem.b . . . . . . . . . . . 12 (𝜑𝐵𝐴)
12 fseqenlem.f . . . . . . . . . . . 12 (𝜑𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)
13 fseqenlem.g . . . . . . . . . . . 12 𝐺 = seq𝜔((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))), {⟨∅, 𝐵⟩})
1410, 11, 12, 13fseqenlem1 9037 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ω) → (𝐺𝑘):(𝐴𝑚 𝑘)–1-1𝐴)
1514adantrr 755 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → (𝐺𝑘):(𝐴𝑚 𝑘)–1-1𝐴)
16 f1f 6262 . . . . . . . . . 10 ((𝐺𝑘):(𝐴𝑚 𝑘)–1-1𝐴 → (𝐺𝑘):(𝐴𝑚 𝑘)⟶𝐴)
1715, 16syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → (𝐺𝑘):(𝐴𝑚 𝑘)⟶𝐴)
18 simprr 813 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → 𝑦 ∈ (𝐴𝑚 𝑘))
1917, 18ffvelrnd 6523 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → ((𝐺𝑘)‘𝑦) ∈ 𝐴)
209, 19eqeltrd 2839 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴)
21 opelxpi 5305 . . . . . . 7 ((dom 𝑦 ∈ ω ∧ ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
227, 20, 21syl2anc 696 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
2322rexlimdvaa 3170 . . . . 5 (𝜑 → (∃𝑘 ∈ ω 𝑦 ∈ (𝐴𝑚 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
241, 23syl5bi 232 . . . 4 (𝜑 → (𝑦 𝑘 ∈ ω (𝐴𝑚 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
2524imp 444 . . 3 ((𝜑𝑦 𝑘 ∈ ω (𝐴𝑚 𝑘)) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
26 fseqenlem.k . . 3 𝐾 = (𝑦 𝑘 ∈ ω (𝐴𝑚 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)
2725, 26fmptd 6548 . 2 (𝜑𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)⟶(ω × 𝐴))
28 ffun 6209 . . . . . . . . . . . . . . 15 (𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)⟶(ω × 𝐴) → Fun 𝐾)
29 funbrfv2b 6402 . . . . . . . . . . . . . . 15 (Fun 𝐾 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾𝑧) = 𝑤)))
3027, 28, 293syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾𝑧) = 𝑤)))
3130simplbda 655 . . . . . . . . . . . . 13 ((𝜑𝑧𝐾𝑤) → (𝐾𝑧) = 𝑤)
3230simprbda 654 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐾𝑤) → 𝑧 ∈ dom 𝐾)
33 fdm 6212 . . . . . . . . . . . . . . . . 17 (𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)⟶(ω × 𝐴) → dom 𝐾 = 𝑘 ∈ ω (𝐴𝑚 𝑘))
3427, 33syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐾 = 𝑘 ∈ ω (𝐴𝑚 𝑘))
3534adantr 472 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐾𝑤) → dom 𝐾 = 𝑘 ∈ ω (𝐴𝑚 𝑘))
3632, 35eleqtrd 2841 . . . . . . . . . . . . . 14 ((𝜑𝑧𝐾𝑤) → 𝑧 𝑘 ∈ ω (𝐴𝑚 𝑘))
37 dmeq 5479 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → dom 𝑦 = dom 𝑧)
3837fveq2d 6356 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝐺‘dom 𝑦) = (𝐺‘dom 𝑧))
39 id 22 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧𝑦 = 𝑧)
4038, 39fveq12d 6358 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘dom 𝑧)‘𝑧))
4137, 40opeq12d 4561 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
42 opex 5081 . . . . . . . . . . . . . . 15 ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩ ∈ V
4341, 26, 42fvmpt 6444 . . . . . . . . . . . . . 14 (𝑧 𝑘 ∈ ω (𝐴𝑚 𝑘) → (𝐾𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4436, 43syl 17 . . . . . . . . . . . . 13 ((𝜑𝑧𝐾𝑤) → (𝐾𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4531, 44eqtr3d 2796 . . . . . . . . . . . 12 ((𝜑𝑧𝐾𝑤) → 𝑤 = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4645fveq2d 6356 . . . . . . . . . . 11 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
47 vex 3343 . . . . . . . . . . . . 13 𝑧 ∈ V
4847dmex 7264 . . . . . . . . . . . 12 dom 𝑧 ∈ V
49 fvex 6362 . . . . . . . . . . . 12 ((𝐺‘dom 𝑧)‘𝑧) ∈ V
5048, 49op1st 7341 . . . . . . . . . . 11 (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = dom 𝑧
5146, 50syl6eq 2810 . . . . . . . . . 10 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = dom 𝑧)
5251fveq2d 6356 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5352cnveqd 5453 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5445fveq2d 6356 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
5548, 49op2nd 7342 . . . . . . . . 9 (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = ((𝐺‘dom 𝑧)‘𝑧)
5654, 55syl6eq 2810 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = ((𝐺‘dom 𝑧)‘𝑧))
5753, 56fveq12d 6358 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘(1st𝑤))‘(2nd𝑤)) = ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)))
58 eliun 4676 . . . . . . . . . . . . 13 (𝑧 𝑘 ∈ ω (𝐴𝑚 𝑘) ↔ ∃𝑘 ∈ ω 𝑧 ∈ (𝐴𝑚 𝑘))
59 elmapi 8045 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝐴𝑚 𝑘) → 𝑧:𝑘𝐴)
6059adantl 473 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → 𝑧:𝑘𝐴)
61 fdm 6212 . . . . . . . . . . . . . . . . 17 (𝑧:𝑘𝐴 → dom 𝑧 = 𝑘)
6260, 61syl 17 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → dom 𝑧 = 𝑘)
63 simpl 474 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → 𝑘 ∈ ω)
6462, 63eqeltrd 2839 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → dom 𝑧 ∈ ω)
65 simpr 479 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → 𝑧 ∈ (𝐴𝑚 𝑘))
6662oveq2d 6829 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → (𝐴𝑚 dom 𝑧) = (𝐴𝑚 𝑘))
6765, 66eleqtrrd 2842 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → 𝑧 ∈ (𝐴𝑚 dom 𝑧))
6864, 67jca 555 . . . . . . . . . . . . . 14 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 dom 𝑧)))
6968rexlimiva 3166 . . . . . . . . . . . . 13 (∃𝑘 ∈ ω 𝑧 ∈ (𝐴𝑚 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 dom 𝑧)))
7058, 69sylbi 207 . . . . . . . . . . . 12 (𝑧 𝑘 ∈ ω (𝐴𝑚 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 dom 𝑧)))
7136, 70syl 17 . . . . . . . . . . 11 ((𝜑𝑧𝐾𝑤) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 dom 𝑧)))
7271simpld 477 . . . . . . . . . 10 ((𝜑𝑧𝐾𝑤) → dom 𝑧 ∈ ω)
7310, 11, 12, 13fseqenlem1 9037 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑧 ∈ ω) → (𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1𝐴)
7472, 73syldan 488 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1𝐴)
75 f1f1orn 6309 . . . . . . . . 9 ((𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1𝐴 → (𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧))
7674, 75syl 17 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧))
7771simprd 482 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → 𝑧 ∈ (𝐴𝑚 dom 𝑧))
78 f1ocnvfv1 6695 . . . . . . . 8 (((𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧) ∧ 𝑧 ∈ (𝐴𝑚 dom 𝑧)) → ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
7976, 77, 78syl2anc 696 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
8057, 79eqtr2d 2795 . . . . . 6 ((𝜑𝑧𝐾𝑤) → 𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤)))
8180ex 449 . . . . 5 (𝜑 → (𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
8281alrimiv 2004 . . . 4 (𝜑 → ∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
83 mo2icl 3526 . . . 4 (∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))) → ∃*𝑧 𝑧𝐾𝑤)
8482, 83syl 17 . . 3 (𝜑 → ∃*𝑧 𝑧𝐾𝑤)
8584alrimiv 2004 . 2 (𝜑 → ∀𝑤∃*𝑧 𝑧𝐾𝑤)
86 dff12 6261 . 2 (𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)–1-1→(ω × 𝐴) ↔ (𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)⟶(ω × 𝐴) ∧ ∀𝑤∃*𝑧 𝑧𝐾𝑤))
8727, 85, 86sylanbrc 701 1 (𝜑𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)–1-1→(ω × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wcel 2139  ∃*wmo 2608  wrex 3051  Vcvv 3340  c0 4058  {csn 4321  cop 4327   ciun 4672   class class class wbr 4804  cmpt 4881   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267  cres 5268  suc csuc 5886  Fun wfun 6043  wf 6045  1-1wf1 6046  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  cmpt2 6815  ωcom 7230  1st c1st 7331  2nd c2nd 7332  seq𝜔cseqom 7711  𝑚 cmap 8023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-seqom 7712  df-1o 7729  df-map 8025
This theorem is referenced by:  fseqen  9040  pwfseqlem5  9677
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