Theorem axcc2 10357

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 2902	. . 3 ⊢ Ⅎ𝑛if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚))
2		nfcv 2902	. . 3 ⊢ Ⅎ𝑚if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛))
3		fveqeq2 6843	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) = ∅ ↔ (𝐹‘𝑛) = ∅))
4		fveq2 6834	. . . 4 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
5	3, 4	ifbieq2d 4488	. . 3 ⊢ (𝑚 = 𝑛 → if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)) = if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
6	1, 2, 5	cbvmpt 5181	. 2 ⊢ (𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚))) = (𝑛 ∈ ω ↦ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
7		nfcv 2902	. . 3 ⊢ Ⅎ𝑛({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚))
8		nfcv 2902	. . . 4 ⊢ Ⅎ𝑚{𝑛}
9		nffvmpt1 6845	. . . 4 ⊢ Ⅎ𝑚((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛)
10	8, 9	nfxp 5658	. . 3 ⊢ Ⅎ𝑚({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛))
11		sneq 4572	. . . 4 ⊢ (𝑚 = 𝑛 → {𝑚} = {𝑛})
12		fveq2 6834	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚) = ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛))
13	11, 12	xpeq12d 5656	. . 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 5181	. 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 2902	. . 3 ⊢ Ⅎ𝑛(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑚)))
16		nfcv 2902	. . . 4 ⊢ Ⅎ𝑚2nd
17		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑚𝑓
18		nffvmpt1 6845	. . . . 5 ⊢ Ⅎ𝑚((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛)
19	17, 18	nffv 6844	. . . 4 ⊢ Ⅎ𝑚(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛))
20	16, 19	nffv 6844	. . 3 ⊢ Ⅎ𝑚(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛)))
21		2fveq3 6839	. . . 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 6838	. . 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 5181	. 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 10356	1 ⊢ ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  ∅c0 4268  ifcif 4461  {csn 4562   ↦ cmpt 5160   × cxp 5623   Fn wfn 6487  ‘cfv 6492  ωcom 7813  2^nd c2nd 7937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-inf2 9560  ax-cc 10355

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7814  df-2nd 7939  df-er 8640  df-en 8891

This theorem is referenced by:  axcc3  10358  acncc  10360  domtriomlem  10362