Theorem cc2 7580

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 5669	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
5	4	eleq2d 2302	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑤 ∈ (𝐹‘𝑥) ↔ 𝑤 ∈ (𝐹‘𝑦)))
6	5	exbidv 1874	. . . . 5 ⊢ (𝑥 = 𝑦 → (∃𝑤 𝑤 ∈ (𝐹‘𝑥) ↔ ∃𝑤 𝑤 ∈ (𝐹‘𝑦)))
7	6	cbvralv 2777	. . . 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 2293	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ (𝐹‘𝑦) ↔ 𝑣 ∈ (𝐹‘𝑦)))
10	9	cbvexv 1968	. . . 4 ⊢ (∃𝑤 𝑤 ∈ (𝐹‘𝑦) ↔ ∃𝑣 𝑣 ∈ (𝐹‘𝑦))
11	10	ralbii 2548	. . 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 2384	. . 3 ⊢ Ⅎ𝑛({𝑚} × (𝐹‘𝑚))
14		nfcv 2384	. . 3 ⊢ Ⅎ𝑚({𝑛} × (𝐹‘𝑛))
15		sneq 3699	. . . 4 ⊢ (𝑚 = 𝑛 → {𝑚} = {𝑛})
16		fveq2 5669	. . . 4 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
17	15, 16	xpeq12d 4773	. . 3 ⊢ (𝑚 = 𝑛 → ({𝑚} × (𝐹‘𝑚)) = ({𝑛} × (𝐹‘𝑛)))
18	13, 14, 17	cbvmpt 4204	. 2 ⊢ (𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚))) = (𝑛 ∈ ω ↦ ({𝑛} × (𝐹‘𝑛)))
19		nfcv 2384	. . 3 ⊢ Ⅎ𝑛(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑚)))
20		nfcv 2384	. . . 4 ⊢ Ⅎ𝑚2nd
21		nfcv 2384	. . . . 5 ⊢ Ⅎ𝑚𝑓
22		nffvmpt1 5680	. . . . 5 ⊢ Ⅎ𝑚((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛)
23	21, 22	nffv 5679	. . . 4 ⊢ Ⅎ𝑚(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛))
24	20, 23	nffv 5679	. . 3 ⊢ Ⅎ𝑚(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛)))
25		2fveq3 5674	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑚)) = (𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛)))
26	25	fveq2d 5673	. . 3 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑚))) = (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹‘𝑚)))‘𝑛))))
27	19, 24, 26	cbvmpt 4204	. 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 7579	1 ⊢ (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛)))

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1541   ∈ wcel 2203  ∀wral 2520  {csn 3688   ↦ cmpt 4170  ωcom 4711   × cxp 4746   Fn wfn 5346  ‘cfv 5351  2^nd c2nd 6332  CCHOICEwacc 7575

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-2nd 6334  df-er 6766  df-en 6975  df-cc 7576

This theorem is referenced by:  cc3  7581  acnccim  7585