Theorem axcc2 10519

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 2922	. . 3 ⊢ Ⅎ𝑛if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚))
2		nfcv 2922	. . 3 ⊢ Ⅎ𝑚if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛))
3		fveqeq2 6895	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) = ∅ ↔ (𝐹‘𝑛) = ∅))
4		fveq2 6886	. . . 4 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
5	3, 4	ifbieq2d 4508	. . 3 ⊢ (𝑚 = 𝑛 → if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)) = if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
6	1, 2, 5	cbvmpt 5206	. 2 ⊢ (𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚))) = (𝑛 ∈ ω ↦ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
7		nfcv 2922	. . 3 ⊢ Ⅎ𝑛({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚))
8		nfcv 2922	. . . 4 ⊢ Ⅎ𝑚{𝑛}
9		nffvmpt1 6897	. . . 4 ⊢ Ⅎ𝑚((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛)
10	8, 9	nfxp 5683	. . 3 ⊢ Ⅎ𝑚({𝑛} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛))
11		sneq 4590	. . . 4 ⊢ (𝑚 = 𝑛 → {𝑚} = {𝑛})
12		fveq2 6886	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚) = ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑛))
13	11, 12	xpeq12d 5681	. . 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 5206	. 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 2922	. . 3 ⊢ Ⅎ𝑛(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑚)))
16		nfcv 2922	. . . 4 ⊢ Ⅎ𝑚2nd
17		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑚𝑓
18		nffvmpt1 6897	. . . . 5 ⊢ Ⅎ𝑚((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛)
19	17, 18	nffv 6896	. . . 4 ⊢ Ⅎ𝑚(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛))
20	16, 19	nffv 6896	. . 3 ⊢ Ⅎ𝑚(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × ((𝑚 ∈ ω ↦ if((𝐹‘𝑚) = ∅, {∅}, (𝐹‘𝑚)))‘𝑚)))‘𝑛)))
21		2fveq3 6891	. . . 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 6890	. . 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 5206	. 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 10518	1 ⊢ ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∅c0 4278  ifcif 4481  {csn 4584   ↦ cmpt 5185   × cxp 5648   Fn wfn 6534  ‘cfv 6539  ωcom 7878  2^nd c2nd 8001

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-inf2 9646  ax-cc 10517

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-om 7879  df-2nd 8003  df-er 8721  df-en 8978

This theorem is used by:  axcc3  10520  acncc  10522  domtriomlem  10524