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Theorem axdclem2 9327
Description: Lemma for axdc 9328. Using the full Axiom of Choice, we can construct a choice function 𝑔 on 𝒫 dom 𝑥. From this, we can build a sequence 𝐹 starting at any value 𝑠 ∈ dom 𝑥 by repeatedly applying 𝑔 to the set (𝐹𝑥) (where 𝑥 is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.)
Hypothesis
Ref Expression
axdclem2.1 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)
Assertion
Ref Expression
axdclem2 (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
Distinct variable groups:   𝑓,𝐹,𝑛   𝑦,𝐹,𝑧,𝑛   𝑓,𝑔,𝑥,𝑛   𝑔,𝑠,𝑦,𝑛   𝑧,𝑔   𝑥,𝑦,𝑧
Allowed substitution hints:   𝐹(𝑥,𝑔,𝑠)

Proof of Theorem axdclem2
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 frfnom 7515 . . . . . . . 8 (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω) Fn ω
2 axdclem2.1 . . . . . . . . 9 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)
32fneq1i 5973 . . . . . . . 8 (𝐹 Fn ω ↔ (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω) Fn ω)
41, 3mpbir 221 . . . . . . 7 𝐹 Fn ω
54a1i 11 . . . . . 6 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → 𝐹 Fn ω)
6 fveq2 6178 . . . . . . . . . . 11 (𝑛 = ∅ → (𝐹𝑛) = (𝐹‘∅))
7 suceq 5778 . . . . . . . . . . . 12 (𝑛 = ∅ → suc 𝑛 = suc ∅)
87fveq2d 6182 . . . . . . . . . . 11 (𝑛 = ∅ → (𝐹‘suc 𝑛) = (𝐹‘suc ∅))
96, 8breq12d 4657 . . . . . . . . . 10 (𝑛 = ∅ → ((𝐹𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘∅)𝑥(𝐹‘suc ∅)))
10 fveq2 6178 . . . . . . . . . . 11 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
11 suceq 5778 . . . . . . . . . . . 12 (𝑛 = 𝑘 → suc 𝑛 = suc 𝑘)
1211fveq2d 6182 . . . . . . . . . . 11 (𝑛 = 𝑘 → (𝐹‘suc 𝑛) = (𝐹‘suc 𝑘))
1310, 12breq12d 4657 . . . . . . . . . 10 (𝑛 = 𝑘 → ((𝐹𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹𝑘)𝑥(𝐹‘suc 𝑘)))
14 fveq2 6178 . . . . . . . . . . 11 (𝑛 = suc 𝑘 → (𝐹𝑛) = (𝐹‘suc 𝑘))
15 suceq 5778 . . . . . . . . . . . 12 (𝑛 = suc 𝑘 → suc 𝑛 = suc suc 𝑘)
1615fveq2d 6182 . . . . . . . . . . 11 (𝑛 = suc 𝑘 → (𝐹‘suc 𝑛) = (𝐹‘suc suc 𝑘))
1714, 16breq12d 4657 . . . . . . . . . 10 (𝑛 = suc 𝑘 → ((𝐹𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
182fveq1i 6179 . . . . . . . . . . . . . . . 16 (𝐹‘∅) = ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅)
19 vex 3198 . . . . . . . . . . . . . . . . 17 𝑠 ∈ V
20 fr0g 7516 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ V → ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅) = 𝑠)
2119, 20ax-mp 5 . . . . . . . . . . . . . . . 16 ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅) = 𝑠
2218, 21eqtri 2642 . . . . . . . . . . . . . . 15 (𝐹‘∅) = 𝑠
2322breq1i 4651 . . . . . . . . . . . . . 14 ((𝐹‘∅)𝑥𝑧𝑠𝑥𝑧)
2423biimpri 218 . . . . . . . . . . . . 13 (𝑠𝑥𝑧 → (𝐹‘∅)𝑥𝑧)
2524eximi 1760 . . . . . . . . . . . 12 (∃𝑧 𝑠𝑥𝑧 → ∃𝑧(𝐹‘∅)𝑥𝑧)
26 peano1 7070 . . . . . . . . . . . . 13 ∅ ∈ ω
272axdclem 9326 . . . . . . . . . . . . 13 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘∅)𝑥𝑧) → (∅ ∈ ω → (𝐹‘∅)𝑥(𝐹‘suc ∅)))
2826, 27mpi 20 . . . . . . . . . . . 12 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘∅)𝑥𝑧) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
2925, 28syl3an3 1359 . . . . . . . . . . 11 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧 𝑠𝑥𝑧) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
30293com23 1269 . . . . . . . . . 10 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
31 fvex 6188 . . . . . . . . . . . . . . . . 17 (𝐹𝑘) ∈ V
32 fvex 6188 . . . . . . . . . . . . . . . . 17 (𝐹‘suc 𝑘) ∈ V
3331, 32brelrn 5345 . . . . . . . . . . . . . . . 16 ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘) ∈ ran 𝑥)
34 ssel 3589 . . . . . . . . . . . . . . . 16 (ran 𝑥 ⊆ dom 𝑥 → ((𝐹‘suc 𝑘) ∈ ran 𝑥 → (𝐹‘suc 𝑘) ∈ dom 𝑥))
3533, 34syl5 34 . . . . . . . . . . . . . . 15 (ran 𝑥 ⊆ dom 𝑥 → ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘) ∈ dom 𝑥))
3632eldm 5310 . . . . . . . . . . . . . . 15 ((𝐹‘suc 𝑘) ∈ dom 𝑥 ↔ ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧)
3735, 36syl6ib 241 . . . . . . . . . . . . . 14 (ran 𝑥 ⊆ dom 𝑥 → ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧))
3837ad2antll 764 . . . . . . . . . . . . 13 ((𝑘 ∈ ω ∧ (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥)) → ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧))
39 peano2 7071 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ω → suc 𝑘 ∈ ω)
402axdclem 9326 . . . . . . . . . . . . . . . . 17 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧) → (suc 𝑘 ∈ ω → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
4139, 40syl5 34 . . . . . . . . . . . . . . . 16 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧) → (𝑘 ∈ ω → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
42413expia 1265 . . . . . . . . . . . . . . 15 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥) → (∃𝑧(𝐹‘suc 𝑘)𝑥𝑧 → (𝑘 ∈ ω → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘))))
4342com3r 87 . . . . . . . . . . . . . 14 (𝑘 ∈ ω → ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥) → (∃𝑧(𝐹‘suc 𝑘)𝑥𝑧 → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘))))
4443imp 445 . . . . . . . . . . . . 13 ((𝑘 ∈ ω ∧ (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥)) → (∃𝑧(𝐹‘suc 𝑘)𝑥𝑧 → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
4538, 44syld 47 . . . . . . . . . . . 12 ((𝑘 ∈ ω ∧ (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥)) → ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
46453adantr2 1219 . . . . . . . . . . 11 ((𝑘 ∈ ω ∧ (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)) → ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
4746ex 450 . . . . . . . . . 10 (𝑘 ∈ ω → ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘))))
489, 13, 17, 30, 47finds2 7079 . . . . . . . . 9 (𝑛 ∈ ω → ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → (𝐹𝑛)𝑥(𝐹‘suc 𝑛)))
4948com12 32 . . . . . . . 8 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → (𝑛 ∈ ω → (𝐹𝑛)𝑥(𝐹‘suc 𝑛)))
50 fvex 6188 . . . . . . . . 9 (𝐹𝑛) ∈ V
51 fvex 6188 . . . . . . . . 9 (𝐹‘suc 𝑛) ∈ V
5250, 51breldm 5318 . . . . . . . 8 ((𝐹𝑛)𝑥(𝐹‘suc 𝑛) → (𝐹𝑛) ∈ dom 𝑥)
5349, 52syl6 35 . . . . . . 7 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → (𝑛 ∈ ω → (𝐹𝑛) ∈ dom 𝑥))
5453ralrimiv 2962 . . . . . 6 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∀𝑛 ∈ ω (𝐹𝑛) ∈ dom 𝑥)
55 ffnfv 6374 . . . . . 6 (𝐹:ω⟶dom 𝑥 ↔ (𝐹 Fn ω ∧ ∀𝑛 ∈ ω (𝐹𝑛) ∈ dom 𝑥))
565, 54, 55sylanbrc 697 . . . . 5 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → 𝐹:ω⟶dom 𝑥)
57 omex 8525 . . . . . 6 ω ∈ V
5857a1i 11 . . . . 5 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ω ∈ V)
59 vex 3198 . . . . . . 7 𝑥 ∈ V
6059dmex 7084 . . . . . 6 dom 𝑥 ∈ V
6160a1i 11 . . . . 5 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → dom 𝑥 ∈ V)
62 fex2 7106 . . . . 5 ((𝐹:ω⟶dom 𝑥 ∧ ω ∈ V ∧ dom 𝑥 ∈ V) → 𝐹 ∈ V)
6356, 58, 61, 62syl3anc 1324 . . . 4 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → 𝐹 ∈ V)
6449ralrimiv 2962 . . . 4 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∀𝑛 ∈ ω (𝐹𝑛)𝑥(𝐹‘suc 𝑛))
65 fveq1 6177 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑛) = (𝐹𝑛))
66 fveq1 6177 . . . . . . 7 (𝑓 = 𝐹 → (𝑓‘suc 𝑛) = (𝐹‘suc 𝑛))
6765, 66breq12d 4657 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝐹𝑛)𝑥(𝐹‘suc 𝑛)))
6867ralbidv 2983 . . . . 5 (𝑓 = 𝐹 → (∀𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝐹𝑛)𝑥(𝐹‘suc 𝑛)))
6968spcegv 3289 . . . 4 (𝐹 ∈ V → (∀𝑛 ∈ ω (𝐹𝑛)𝑥(𝐹‘suc 𝑛) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
7063, 64, 69sylc 65 . . 3 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
71703exp 1262 . 2 (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))))
7260pwex 4839 . . 3 𝒫 dom 𝑥 ∈ V
7372ac4c 9283 . 2 𝑔𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)
7471, 73exlimiiv 1857 1 (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wne 2791  wral 2909  Vcvv 3195  wss 3567  c0 3907  𝒫 cpw 4149   class class class wbr 4644  cmpt 4720  dom cdm 5104  ran crn 5105  cres 5106  suc csuc 5713   Fn wfn 5871  wf 5872  cfv 5876  ωcom 7050  reccrdg 7490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-ac2 9270
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-ac 8924
This theorem is referenced by:  axdc  9328
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