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Theorem axcc2 9461
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 2913 . . 3 𝑛if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚))
2 nfcv 2913 . . 3 𝑚if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛))
3 fveq2 6332 . . . . 5 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
43eqeq1d 2773 . . . 4 (𝑚 = 𝑛 → ((𝐹𝑚) = ∅ ↔ (𝐹𝑛) = ∅))
54, 3ifbieq2d 4250 . . 3 (𝑚 = 𝑛 → if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
61, 2, 5cbvmpt 4883 . 2 (𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚))) = (𝑛 ∈ ω ↦ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
7 nfcv 2913 . . 3 𝑛({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚))
8 nfcv 2913 . . . 4 𝑚{𝑛}
9 nffvmpt1 6340 . . . 4 𝑚((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)
108, 9nfxp 5282 . . 3 𝑚({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛))
11 sneq 4326 . . . 4 (𝑚 = 𝑛 → {𝑚} = {𝑛})
12 fveq2 6332 . . . 4 (𝑚 = 𝑛 → ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚) = ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛))
1311, 12xpeq12d 5280 . . 3 (𝑚 = 𝑛 → ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)) = ({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)))
147, 10, 13cbvmpt 4883 . 2 (𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚))) = (𝑛 ∈ ω ↦ ({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)))
15 nfcv 2913 . . 3 𝑛(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑚)))
16 nfcv 2913 . . . 4 𝑚2nd
17 nfcv 2913 . . . . 5 𝑚𝑓
18 nffvmpt1 6340 . . . . 5 𝑚((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛)
1917, 18nffv 6339 . . . 4 𝑚(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛))
2016, 19nffv 6339 . . 3 𝑚(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛)))
21 fveq2 6332 . . . . 5 (𝑚 = 𝑛 → ((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑚) = ((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛))
2221fveq2d 6336 . . . 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 6336 . . 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 4883 . 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 9460 1 𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  c0 4063  ifcif 4225  {csn 4316  cmpt 4863   × cxp 5247   Fn wfn 6026  cfv 6031  ωcom 7212  2nd c2nd 7314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cc 9459
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-2nd 7316  df-er 7896  df-en 8110
This theorem is referenced by:  axcc3  9462  acncc  9464  domtriomlem  9466
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