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Theorem axcc2 9540
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 2948 . . 3 𝑛if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚))
2 nfcv 2948 . . 3 𝑚if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛))
3 fveqeq2 6413 . . . 4 (𝑚 = 𝑛 → ((𝐹𝑚) = ∅ ↔ (𝐹𝑛) = ∅))
4 fveq2 6404 . . . 4 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
53, 4ifbieq2d 4304 . . 3 (𝑚 = 𝑛 → if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
61, 2, 5cbvmpt 4943 . 2 (𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚))) = (𝑛 ∈ ω ↦ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
7 nfcv 2948 . . 3 𝑛({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚))
8 nfcv 2948 . . . 4 𝑚{𝑛}
9 nffvmpt1 6415 . . . 4 𝑚((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)
108, 9nfxp 5343 . . 3 𝑚({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛))
11 sneq 4380 . . . 4 (𝑚 = 𝑛 → {𝑚} = {𝑛})
12 fveq2 6404 . . . 4 (𝑚 = 𝑛 → ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚) = ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛))
1311, 12xpeq12d 5341 . . 3 (𝑚 = 𝑛 → ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)) = ({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)))
147, 10, 13cbvmpt 4943 . 2 (𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚))) = (𝑛 ∈ ω ↦ ({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)))
15 nfcv 2948 . . 3 𝑛(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑚)))
16 nfcv 2948 . . . 4 𝑚2nd
17 nfcv 2948 . . . . 5 𝑚𝑓
18 nffvmpt1 6415 . . . . 5 𝑚((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛)
1917, 18nffv 6414 . . . 4 𝑚(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛))
2016, 19nffv 6414 . . 3 𝑚(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛)))
21 2fveq3 6409 . . . 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 6408 . . 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 4943 . 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 9539 1 𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wex 1859  wcel 2156  wne 2978  wral 3096  c0 4116  ifcif 4279  {csn 4370  cmpt 4923   × cxp 5309   Fn wfn 6092  cfv 6097  ωcom 7291  2nd c2nd 7393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-inf2 8781  ax-cc 9538
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-om 7292  df-2nd 7395  df-er 7975  df-en 8189
This theorem is referenced by:  axcc3  9541  acncc  9543  domtriomlem  9545
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