Theorem axcc2 10506

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 ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛)))

Distinct variable group:   𝑔,𝐹,𝑛

Proof of Theorem axcc2

Dummy variables 𝑓 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2908	. . 3 ⊢ Ⅎ𝑛if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚))
2		nfcv 2908	. . 3 ⊢ Ⅎ𝑚if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛))
3		fveqeq2 6929	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) = ∅ ↔ (𝐹‘𝑛) = ∅))
4		fveq2 6920	. . . 4 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
5	3, 4	ifbieq2d 4574	. . 3 ⊢ (𝑚 = 𝑛 → if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)) = if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
6	1, 2, 5	cbvmpt 5277	. 2 ⊢ (𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚))) = (𝑛 ∈ ω ↦ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
7		nfcv 2908	. . 3 ⊢ Ⅎ𝑛({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚))
8		nfcv 2908	. . . 4 ⊢ Ⅎ𝑚{𝑛}
9		nffvmpt1 6931	. . . 4 ⊢ Ⅎ𝑚((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛)
10	8, 9	nfxp 5733	. . 3 ⊢ Ⅎ𝑚({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛))
11		sneq 4658	. . . 4 ⊢ (𝑚 = 𝑛 → {𝑚} = {𝑛})
12		fveq2 6920	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚) = ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛))
13	11, 12	xpeq12d 5731	. . 3 ⊢ (𝑚 = 𝑛 → ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)) = ({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛)))
14	7, 10, 13	cbvmpt 5277	. 2 ⊢ (𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚))) = (𝑛 ∈ ω ↦ ({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛)))
15		nfcv 2908	. . 3 ⊢ Ⅎ𝑛(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑚)))
16		nfcv 2908	. . . 4 ⊢ Ⅎ𝑚2nd
17		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑚𝑓
18		nffvmpt1 6931	. . . . 5 ⊢ Ⅎ𝑚((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛)
19	17, 18	nffv 6930	. . . 4 ⊢ Ⅎ𝑚(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛))
20	16, 19	nffv 6930	. . 3 ⊢ Ⅎ𝑚(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛)))
21		2fveq3 6925	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑚)) = (𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛)))
22	21	fveq2d 6924	. . 3 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑚))) = (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛))))
23	15, 20, 22	cbvmpt 5277	. 2 ⊢ (𝑚 ∈ ω ↦ (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑚)))) = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛))))
24	6, 14, 23	axcc2lem 10505	1 ⊢ ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∅c0 4352  ifcif 4548  {csn 4648   ↦ cmpt 5249   × cxp 5698   Fn wfn 6568  ‘cfv 6573  ωcom 7903  2^nd c2nd 8029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cc 10504

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-2nd 8031  df-er 8763  df-en 9004

This theorem is referenced by:  axcc3  10507  acncc  10509  domtriomlem  10511