Theorem cc2 7640

Description: Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.)

Hypotheses

Ref	Expression
cc2.cc	⊢ (𝜑 → CCHOICE)
cc2.a	⊢ (𝜑 → 𝐹 Fn ω)
cc2.m	⊢ (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑥))

Assertion

Ref	Expression
cc2	⊢ (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛)))

Distinct variable groups:   𝑔,𝐹,𝑛   𝑤,𝐹,𝑥   𝜑,𝑛

Allowed substitution hints:   𝜑(𝑥, 𝑤, 𝑔)

Proof of Theorem cc2

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

Step	Hyp	Ref	Expression
1		cc2.cc	. 2 ⊢ (𝜑 → CCHOICE)
2		cc2.a	. 2 ⊢ (𝜑 → 𝐹 Fn ω)
3		cc2.m	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑥))
4		fveq2 5698	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
5	4	eleq2d 2308	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑤 ∈ (𝐹‘𝑥) ↔ 𝑤 ∈ (𝐹‘𝑦)))
6	5	exbidv 1878	. . . . 5 ⊢ (𝑥 = 𝑦 → (∃𝑤 𝑤 ∈ (𝐹‘𝑥) ↔ ∃𝑤 𝑤 ∈ (𝐹‘𝑦)))
7	6	cbvralv 2786	. . . 4 ⊢ (∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑥) ↔ ∀𝑦 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑦))
8	3, 7	sylib 122	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑦))
9		eleq1w 2299	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ (𝐹‘𝑦) ↔ 𝑣 ∈ (𝐹‘𝑦)))
10	9	cbvexv 1974	. . . 4 ⊢ (∃𝑤 𝑤 ∈ (𝐹‘𝑦) ↔ ∃𝑣 𝑣 ∈ (𝐹‘𝑦))
11	10	ralbii 2556	. . 3 ⊢ (∀𝑦 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑦) ↔ ∀𝑦 ∈ ω ∃𝑣 𝑣 ∈ (𝐹‘𝑦))
12	8, 11	sylib 122	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ω ∃𝑣 𝑣 ∈ (𝐹‘𝑦))
13		nfcv 2392	. . 3 ⊢ Ⅎ𝑛({𝑚} × (𝐹‘𝑚))
14		nfcv 2392	. . 3 ⊢ Ⅎ𝑚({𝑛} × (𝐹‘𝑛))
15		sneq 3720	. . . 4 ⊢ (𝑚 = 𝑛 → {𝑚} = {𝑛})
16		fveq2 5698	. . . 4 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
17	15, 16	xpeq12d 4801	. . 3 ⊢ (𝑚 = 𝑛 → ({𝑚} × (𝐹‘𝑚)) = ({𝑛} × (𝐹‘𝑛)))
18	13, 14, 17	cbvmpt 4228	. 2 ⊢ (𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚))) = (𝑛 ∈ ω ↦ ({𝑛} × (𝐹‘𝑛)))
19		nfcv 2392	. . 3 ⊢ Ⅎ𝑛(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑚)))
20		nfcv 2392	. . . 4 ⊢ Ⅎ𝑚2nd
21		nfcv 2392	. . . . 5 ⊢ Ⅎ𝑚𝑓
22		nffvmpt1 5709	. . . . 5 ⊢ Ⅎ𝑚((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛)
23	21, 22	nffv 5708	. . . 4 ⊢ Ⅎ𝑚(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛))
24	20, 23	nffv 5708	. . 3 ⊢ Ⅎ𝑚(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛)))
25		2fveq3 5703	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑚)) = (𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛)))
26	25	fveq2d 5702	. . 3 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑚))) = (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛))))
27	19, 24, 26	cbvmpt 4228	. 2 ⊢ (𝑚 ∈ ω ↦ (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑚)))) = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛))))
28	1, 2, 12, 18, 27	cc2lem 7639	1 ⊢ (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  {csn 3709   ↦ cmpt 4193  ωcom 4739   × cxp 4774   Fn wfn 5375  ‘cfv 5380  2^nd c2nd 6377  CCHOICEwacc 7635

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-2nd 6379  df-er 6811  df-en 7028  df-cc 7636

This theorem is used by:  cc3  7641  acnccim  7645