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Theorem axcc2 10374
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 2908 . . 3 𝑛if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚))
2 nfcv 2908 . . 3 𝑚if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛))
3 fveqeq2 6852 . . . 4 (𝑚 = 𝑛 → ((𝐹𝑚) = ∅ ↔ (𝐹𝑛) = ∅))
4 fveq2 6843 . . . 4 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
53, 4ifbieq2d 4513 . . 3 (𝑚 = 𝑛 → if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
61, 2, 5cbvmpt 5217 . 2 (𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚))) = (𝑛 ∈ ω ↦ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
7 nfcv 2908 . . 3 𝑛({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚))
8 nfcv 2908 . . . 4 𝑚{𝑛}
9 nffvmpt1 6854 . . . 4 𝑚((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)
108, 9nfxp 5667 . . 3 𝑚({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛))
11 sneq 4597 . . . 4 (𝑚 = 𝑛 → {𝑚} = {𝑛})
12 fveq2 6843 . . . 4 (𝑚 = 𝑛 → ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚) = ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛))
1311, 12xpeq12d 5665 . . 3 (𝑚 = 𝑛 → ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)) = ({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)))
147, 10, 13cbvmpt 5217 . 2 (𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚))) = (𝑛 ∈ ω ↦ ({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)))
15 nfcv 2908 . . 3 𝑛(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑚)))
16 nfcv 2908 . . . 4 𝑚2nd
17 nfcv 2908 . . . . 5 𝑚𝑓
18 nffvmpt1 6854 . . . . 5 𝑚((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛)
1917, 18nffv 6853 . . . 4 𝑚(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛))
2016, 19nffv 6853 . . 3 𝑚(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛)))
21 2fveq3 6848 . . . 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)))
2221fveq2d 6847 . . 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))))
2315, 20, 22cbvmpt 5217 . 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))))
246, 14, 23axcc2lem 10373 1 𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2944  wral 3065  c0 4283  ifcif 4487  {csn 4587  cmpt 5189   × cxp 5632   Fn wfn 6492  cfv 6497  ωcom 7803  2nd c2nd 7921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9578  ax-cc 10372
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-om 7804  df-2nd 7923  df-er 8649  df-en 8885
This theorem is referenced by:  axcc3  10375  acncc  10377  domtriomlem  10379
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