Theorem cc2 7461

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 5629	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
5	4	eleq2d 2299	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑤 ∈ (𝐹‘𝑥) ↔ 𝑤 ∈ (𝐹‘𝑦)))
6	5	exbidv 1871	. . . . 5 ⊢ (𝑥 = 𝑦 → (∃𝑤 𝑤 ∈ (𝐹‘𝑥) ↔ ∃𝑤 𝑤 ∈ (𝐹‘𝑦)))
7	6	cbvralv 2765	. . . 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 2290	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ (𝐹‘𝑦) ↔ 𝑣 ∈ (𝐹‘𝑦)))
10	9	cbvexv 1965	. . . 4 ⊢ (∃𝑤 𝑤 ∈ (𝐹‘𝑦) ↔ ∃𝑣 𝑣 ∈ (𝐹‘𝑦))
11	10	ralbii 2536	. . 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 2372	. . 3 ⊢ Ⅎ𝑛({𝑚} × (𝐹‘𝑚))
14		nfcv 2372	. . 3 ⊢ Ⅎ𝑚({𝑛} × (𝐹‘𝑛))
15		sneq 3677	. . . 4 ⊢ (𝑚 = 𝑛 → {𝑚} = {𝑛})
16		fveq2 5629	. . . 4 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
17	15, 16	xpeq12d 4744	. . 3 ⊢ (𝑚 = 𝑛 → ({𝑚} × (𝐹‘𝑚)) = ({𝑛} × (𝐹‘𝑛)))
18	13, 14, 17	cbvmpt 4179	. 2 ⊢ (𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚))) = (𝑛 ∈ ω ↦ ({𝑛} × (𝐹‘𝑛)))
19		nfcv 2372	. . 3 ⊢ Ⅎ𝑛(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑚)))
20		nfcv 2372	. . . 4 ⊢ Ⅎ𝑚2nd
21		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑚𝑓
22		nffvmpt1 5640	. . . . 5 ⊢ Ⅎ𝑚((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛)
23	21, 22	nffv 5639	. . . 4 ⊢ Ⅎ𝑚(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛))
24	20, 23	nffv 5639	. . 3 ⊢ Ⅎ𝑚(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛)))
25		2fveq3 5634	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑚)) = (𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛)))
26	25	fveq2d 5633	. . 3 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑚))) = (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛))))
27	19, 24, 26	cbvmpt 4179	. 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 7460	1 ⊢ (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛)))

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  {csn 3666   ↦ cmpt 4145  ωcom 4682   × cxp 4717   Fn wfn 5313  ‘cfv 5318  2^nd c2nd 6291  CCHOICEwacc 7456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-2nd 6293  df-er 6688  df-en 6896  df-cc 7457

This theorem is referenced by:  cc3  7462  acnccim  7466