Theorem axcc2 10359

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 6849	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) = ∅ ↔ (𝐹‘𝑛) = ∅))
4		fveq2 6840	. . . 4 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
5	3, 4	ifbieq2d 4493	. . 3 ⊢ (𝑚 = 𝑛 → if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)) = if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
6	1, 2, 5	cbvmpt 5187	. 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 6851	. . . 4 ⊢ Ⅎ𝑚((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛)
10	8, 9	nfxp 5664	. . 3 ⊢ Ⅎ𝑚({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛))
11		sneq 4577	. . . 4 ⊢ (𝑚 = 𝑛 → {𝑚} = {𝑛})
12		fveq2 6840	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚) = ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛))
13	11, 12	xpeq12d 5662	. . 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 5187	. 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 6851	. . . . 5 ⊢ Ⅎ𝑚((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛)
19	17, 18	nffv 6850	. . . 4 ⊢ Ⅎ𝑚(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛))
20	16, 19	nffv 6850	. . 3 ⊢ Ⅎ𝑚(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛)))
21		2fveq3 6845	. . . 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 6844	. . 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 5187	. 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 10358	1 ⊢ ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∅c0 4273  ifcif 4466  {csn 4567   ↦ cmpt 5166   × cxp 5629   Fn wfn 6493  ‘cfv 6498  ωcom 7817  2^nd c2nd 7941

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-cc 10357

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-om 7818  df-2nd 7943  df-er 8643  df-en 8894

This theorem is referenced by:  axcc3  10360  acncc  10362  domtriomlem  10364