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Theorem axcc2 9203
Description: A possibly more useful version of ax-cc using sequences instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.)
Assertion
Ref Expression
axcc2 𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))
Distinct variable group:   𝑔,𝐹,𝑛

Proof of Theorem axcc2
Dummy variables 𝑓 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2761 . . 3 𝑛if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚))
2 nfcv 2761 . . 3 𝑚if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛))
3 fveq2 6148 . . . . 5 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
43eqeq1d 2623 . . . 4 (𝑚 = 𝑛 → ((𝐹𝑚) = ∅ ↔ (𝐹𝑛) = ∅))
54, 3ifbieq2d 4083 . . 3 (𝑚 = 𝑛 → if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
61, 2, 5cbvmpt 4709 . 2 (𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚))) = (𝑛 ∈ ω ↦ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
7 nfcv 2761 . . 3 𝑛({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚))
8 nfcv 2761 . . . 4 𝑚{𝑛}
9 nffvmpt1 6156 . . . 4 𝑚((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)
108, 9nfxp 5102 . . 3 𝑚({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛))
11 sneq 4158 . . . 4 (𝑚 = 𝑛 → {𝑚} = {𝑛})
12 fveq2 6148 . . . 4 (𝑚 = 𝑛 → ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚) = ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛))
1311, 12xpeq12d 5100 . . 3 (𝑚 = 𝑛 → ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)) = ({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)))
147, 10, 13cbvmpt 4709 . 2 (𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚))) = (𝑛 ∈ ω ↦ ({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)))
15 nfcv 2761 . . 3 𝑛(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑚)))
16 nfcv 2761 . . . 4 𝑚2nd
17 nfcv 2761 . . . . 5 𝑚𝑓
18 nffvmpt1 6156 . . . . 5 𝑚((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛)
1917, 18nffv 6155 . . . 4 𝑚(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛))
2016, 19nffv 6155 . . 3 𝑚(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛)))
21 fveq2 6148 . . . . 5 (𝑚 = 𝑛 → ((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑚) = ((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛))
2221fveq2d 6152 . . . 4 (𝑚 = 𝑛 → (𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑚)) = (𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛)))
2322fveq2d 6152 . . 3 (𝑚 = 𝑛 → (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑚))) = (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛))))
2415, 20, 23cbvmpt 4709 . 2 (𝑚 ∈ ω ↦ (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑚)))) = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛))))
256, 14, 24axcc2lem 9202 1 𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  c0 3891  ifcif 4058  {csn 4148  cmpt 4673   × cxp 5072   Fn wfn 5842  cfv 5847  ωcom 7012  2nd c2nd 7112
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-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cc 9201
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 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-2nd 7114  df-er 7687  df-en 7900
This theorem is referenced by:  axcc3  9204  acncc  9206  domtriomlem  9208
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