Theorem axcc2 10347

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 2898	. . 3 ⊢ Ⅎ𝑛if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚))
2		nfcv 2898	. . 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 4506	. . 3 ⊢ (𝑚 = 𝑛 → if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)) = if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
6	1, 2, 5	cbvmpt 5200	. 2 ⊢ (𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚))) = (𝑛 ∈ ω ↦ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
7		nfcv 2898	. . 3 ⊢ Ⅎ𝑛({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚))
8		nfcv 2898	. . . 4 ⊢ Ⅎ𝑚{𝑛}
9		nffvmpt1 6845	. . . 4 ⊢ Ⅎ𝑚((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛)
10	8, 9	nfxp 5657	. . 3 ⊢ Ⅎ𝑚({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛))
11		sneq 4590	. . . 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 5655	. . 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 5200	. 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 2898	. . 3 ⊢ Ⅎ𝑛(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑚)))
16		nfcv 2898	. . . 4 ⊢ Ⅎ𝑚2nd
17		nfcv 2898	. . . . 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 5200	. 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 10346	1 ⊢ ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∅c0 4285  ifcif 4479  {csn 4580   ↦ cmpt 5179   × cxp 5622   Fn wfn 6487  ‘cfv 6492  ωcom 7808  2^nd c2nd 7932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cc 10345

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7809  df-2nd 7934  df-er 8635  df-en 8884

This theorem is referenced by:  axcc3  10348  acncc  10350  domtriomlem  10352