Theorem cc2 7216

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 5494	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
5	4	eleq2d 2240	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑤 ∈ (𝐹‘𝑥) ↔ 𝑤 ∈ (𝐹‘𝑦)))
6	5	exbidv 1818	. . . . 5 ⊢ (𝑥 = 𝑦 → (∃𝑤 𝑤 ∈ (𝐹‘𝑥) ↔ ∃𝑤 𝑤 ∈ (𝐹‘𝑦)))
7	6	cbvralv 2696	. . . 4 ⊢ (∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑥) ↔ ∀𝑦 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑦))
8	3, 7	sylib 121	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑦))
9		eleq1w 2231	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ (𝐹‘𝑦) ↔ 𝑣 ∈ (𝐹‘𝑦)))
10	9	cbvexv 1911	. . . 4 ⊢ (∃𝑤 𝑤 ∈ (𝐹‘𝑦) ↔ ∃𝑣 𝑣 ∈ (𝐹‘𝑦))
11	10	ralbii 2476	. . 3 ⊢ (∀𝑦 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑦) ↔ ∀𝑦 ∈ ω ∃𝑣 𝑣 ∈ (𝐹‘𝑦))
12	8, 11	sylib 121	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ω ∃𝑣 𝑣 ∈ (𝐹‘𝑦))
13		nfcv 2312	. . 3 ⊢ Ⅎ𝑛({𝑚} × (𝐹‘𝑚))
14		nfcv 2312	. . 3 ⊢ Ⅎ𝑚({𝑛} × (𝐹‘𝑛))
15		sneq 3592	. . . 4 ⊢ (𝑚 = 𝑛 → {𝑚} = {𝑛})
16		fveq2 5494	. . . 4 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
17	15, 16	xpeq12d 4634	. . 3 ⊢ (𝑚 = 𝑛 → ({𝑚} × (𝐹‘𝑚)) = ({𝑛} × (𝐹‘𝑛)))
18	13, 14, 17	cbvmpt 4082	. 2 ⊢ (𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚))) = (𝑛 ∈ ω ↦ ({𝑛} × (𝐹‘𝑛)))
19		nfcv 2312	. . 3 ⊢ Ⅎ𝑛(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑚)))
20		nfcv 2312	. . . 4 ⊢ Ⅎ𝑚2nd
21		nfcv 2312	. . . . 5 ⊢ Ⅎ𝑚𝑓
22		nffvmpt1 5505	. . . . 5 ⊢ Ⅎ𝑚((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛)
23	21, 22	nffv 5504	. . . 4 ⊢ Ⅎ𝑚(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛))
24	20, 23	nffv 5504	. . 3 ⊢ Ⅎ𝑚(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛)))
25		2fveq3 5499	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑚)) = (𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛)))
26	25	fveq2d 5498	. . 3 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑚))) = (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛))))
27	19, 24, 26	cbvmpt 4082	. 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 7215	1 ⊢ (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛)))

Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  {csn 3581   ↦ cmpt 4048  ωcom 4572   × cxp 4607   Fn wfn 5191  ‘cfv 5196  2^nd c2nd 6115  CCHOICEwacc 7211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-2nd 6117  df-er 6509  df-en 6715  df-cc 7212

This theorem is referenced by:  cc3  7217