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Theorem axcc2 9846
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 2980 . . 3 𝑛if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚))
2 nfcv 2980 . . 3 𝑚if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛))
3 fveqeq2 6662 . . . 4 (𝑚 = 𝑛 → ((𝐹𝑚) = ∅ ↔ (𝐹𝑛) = ∅))
4 fveq2 6653 . . . 4 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
53, 4ifbieq2d 4473 . . 3 (𝑚 = 𝑛 → if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
61, 2, 5cbvmpt 5150 . 2 (𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚))) = (𝑛 ∈ ω ↦ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
7 nfcv 2980 . . 3 𝑛({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚))
8 nfcv 2980 . . . 4 𝑚{𝑛}
9 nffvmpt1 6664 . . . 4 𝑚((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)
108, 9nfxp 5571 . . 3 𝑚({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛))
11 sneq 4558 . . . 4 (𝑚 = 𝑛 → {𝑚} = {𝑛})
12 fveq2 6653 . . . 4 (𝑚 = 𝑛 → ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚) = ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛))
1311, 12xpeq12d 5569 . . 3 (𝑚 = 𝑛 → ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)) = ({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)))
147, 10, 13cbvmpt 5150 . 2 (𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚))) = (𝑛 ∈ ω ↦ ({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑛)))
15 nfcv 2980 . . 3 𝑛(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑚)))
16 nfcv 2980 . . . 4 𝑚2nd
17 nfcv 2980 . . . . 5 𝑚𝑓
18 nffvmpt1 6664 . . . . 5 𝑚((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛)
1917, 18nffv 6663 . . . 4 𝑚(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛))
2016, 19nffv 6663 . . 3 𝑚(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹𝑚) = ∅, {∅}, (𝐹𝑚)))‘𝑚)))‘𝑛)))
21 2fveq3 6658 . . . 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 6657 . . 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 5150 . 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 9845 1 𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2115  wne 3013  wral 3132  c0 4274  ifcif 4448  {csn 4548  cmpt 5129   × cxp 5536   Fn wfn 6333  cfv 6338  ωcom 7565  2nd c2nd 7673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446  ax-inf2 9090  ax-cc 9844
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-om 7566  df-2nd 7675  df-er 8274  df-en 8495
This theorem is referenced by:  axcc3  9847  acncc  9849  domtriomlem  9851
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