cc3 - Intuitionistic Logic Explorer

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Theorem cc3 7100

Description: Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.)

**Hypotheses**
Ref	Expression
cc3.cc	⊢ (𝜑 → CCHOICE)
cc3.f	⊢ (𝜑 → ∀𝑛 ∈ 𝑁 𝐹 ∈ V)
cc3.m	⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑤 𝑤 ∈ 𝐹)
cc3.n	⊢ (𝜑 → 𝑁 ≈ ω)

**Assertion**
Ref	Expression
cc3	⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹))

Distinct variable groups: 𝑓,𝐹 𝑤,𝐹 𝑓,𝑁,𝑛 𝑤,𝑁,𝑛 𝜑,𝑛,𝑤 Allowed substitution hints: 𝜑(𝑓) 𝐹(𝑛)

Proof of Theorem cc3 Dummy variables 𝑔 ℎ 𝑘 𝑚 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cc3.n	. . 3 ⊢ (𝜑 → 𝑁 ≈ ω)
2		relen 6646	. . . 4 ⊢ Rel ≈
3	2	brrelex1i 4590	. . 3 ⊢ (𝑁 ≈ ω → 𝑁 ∈ V)
4		mptexg 5653	. . 3 ⊢ (𝑁 ∈ V → (𝑛 ∈ 𝑁 ↦ 𝐹) ∈ V)
5	1, 3, 4	3syl 17	. 2 ⊢ (𝜑 → (𝑛 ∈ 𝑁 ↦ 𝐹) ∈ V)
6		bren 6649	. . . . . . 7 ⊢ (𝑁 ≈ ω ↔ ∃ℎ ℎ:𝑁–1-1-onto→ω)
7	1, 6	sylib 121	. . . . . 6 ⊢ (𝜑 → ∃ℎ ℎ:𝑁–1-1-onto→ω)
8	7	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) → ∃ℎ ℎ:𝑁–1-1-onto→ω)
9		cc3.cc	. . . . . . . 8 ⊢ (𝜑 → CCHOICE)
10	9	ad2antrr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → CCHOICE)
11		cc3.f	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 𝐹 ∈ V)
12		eqid 2140	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑁 ↦ 𝐹) = (𝑛 ∈ 𝑁 ↦ 𝐹)
13	12	mptfng 5256	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ 𝑁 𝐹 ∈ V ↔ (𝑛 ∈ 𝑁 ↦ 𝐹) Fn 𝑁)
14	11, 13	sylib 121	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ 𝑁 ↦ 𝐹) Fn 𝑁)
15	14	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) → (𝑛 ∈ 𝑁 ↦ 𝐹) Fn 𝑁)
16		simpr 109	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) → 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹))
17	16	fneq1d 5221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) → (𝑘 Fn 𝑁 ↔ (𝑛 ∈ 𝑁 ↦ 𝐹) Fn 𝑁))
18	15, 17	mpbird 166	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) → 𝑘 Fn 𝑁)
19	18	adantr 274	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → 𝑘 Fn 𝑁)
20		f1ocnv 5388	. . . . . . . . . 10 ⊢ (ℎ:𝑁–1-1-onto→ω → ^◡ℎ:ω–1-1-onto→𝑁)
21	20	adantl 275	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → ^◡ℎ:ω–1-1-onto→𝑁)
22		f1of 5375	. . . . . . . . 9 ⊢ (^◡ℎ:ω–1-1-onto→𝑁 → ^◡ℎ:ω⟶𝑁)
23	21, 22	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → ^◡ℎ:ω⟶𝑁)
24		fnfco 5305	. . . . . . . 8 ⊢ ((𝑘 Fn 𝑁 ∧ ^◡ℎ:ω⟶𝑁) → (𝑘 ∘ ^◡ℎ) Fn ω)
25	19, 23, 24	syl2anc 409	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → (𝑘 ∘ ^◡ℎ) Fn ω)
26	23	ffvelrnda 5563	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → (^◡ℎ‘𝑝) ∈ 𝑁)
27		cc3.m	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑤 𝑤 ∈ 𝐹)
28	27	ad3antrrr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ∀𝑛 ∈ 𝑁 ∃𝑤 𝑤 ∈ 𝐹)
29		nfcsb1v 3040	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹
30	29	nfcri 2276	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑤 ∈ ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹
31	30	nfex 1617	. . . . . . . . . . 11 ⊢ Ⅎ𝑛∃𝑤 𝑤 ∈ ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹
32		csbeq1a 3016	. . . . . . . . . . . . 13 ⊢ (𝑛 = (^◡ℎ‘𝑝) → 𝐹 = ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹)
33	32	eleq2d 2210	. . . . . . . . . . . 12 ⊢ (𝑛 = (^◡ℎ‘𝑝) → (𝑤 ∈ 𝐹 ↔ 𝑤 ∈ ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹))
34	33	exbidv 1798	. . . . . . . . . . 11 ⊢ (𝑛 = (^◡ℎ‘𝑝) → (∃𝑤 𝑤 ∈ 𝐹 ↔ ∃𝑤 𝑤 ∈ ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹))
35	31, 34	rspc 2787	. . . . . . . . . 10 ⊢ ((^◡ℎ‘𝑝) ∈ 𝑁 → (∀𝑛 ∈ 𝑁 ∃𝑤 𝑤 ∈ 𝐹 → ∃𝑤 𝑤 ∈ ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹))
36	26, 28, 35	sylc 62	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ∃𝑤 𝑤 ∈ ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹)
37		fvco3 5500	. . . . . . . . . . . . 13 ⊢ ((^◡ℎ:ω⟶𝑁 ∧ 𝑝 ∈ ω) → ((𝑘 ∘ ^◡ℎ)‘𝑝) = (𝑘‘(^◡ℎ‘𝑝)))
38	23, 37	sylan 281	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ((𝑘 ∘ ^◡ℎ)‘𝑝) = (𝑘‘(^◡ℎ‘𝑝)))
39		simpllr 524	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹))
40	39	fveq1d 5431	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → (𝑘‘(^◡ℎ‘𝑝)) = ((𝑛 ∈ 𝑁 ↦ 𝐹)‘(^◡ℎ‘𝑝)))
41	11	ad3antrrr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ∀𝑛 ∈ 𝑁 𝐹 ∈ V)
42	29	nfel1 2293	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹 ∈ V
43	32	eleq1d 2209	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (^◡ℎ‘𝑝) → (𝐹 ∈ V ↔ ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹 ∈ V))
44	42, 43	rspc 2787	. . . . . . . . . . . . . . 15 ⊢ ((^◡ℎ‘𝑝) ∈ 𝑁 → (∀𝑛 ∈ 𝑁 𝐹 ∈ V → ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹 ∈ V))
45	26, 41, 44	sylc 62	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹 ∈ V)
46	12	fvmpts 5507	. . . . . . . . . . . . . 14 ⊢ (((^◡ℎ‘𝑝) ∈ 𝑁 ∧ ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹 ∈ V) → ((𝑛 ∈ 𝑁 ↦ 𝐹)‘(^◡ℎ‘𝑝)) = ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹)
47	26, 45, 46	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ((𝑛 ∈ 𝑁 ↦ 𝐹)‘(^◡ℎ‘𝑝)) = ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹)
48	40, 47	eqtrd 2173	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → (𝑘‘(^◡ℎ‘𝑝)) = ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹)
49	38, 48	eqtrd 2173	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ((𝑘 ∘ ^◡ℎ)‘𝑝) = ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹)
50	49	eleq2d 2210	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → (𝑤 ∈ ((𝑘 ∘ ^◡ℎ)‘𝑝) ↔ 𝑤 ∈ ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹))
51	50	exbidv 1798	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → (∃𝑤 𝑤 ∈ ((𝑘 ∘ ^◡ℎ)‘𝑝) ↔ ∃𝑤 𝑤 ∈ ⦋(^◡ℎ‘𝑝) / 𝑛⦌𝐹))
52	36, 51	mpbird 166	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ∃𝑤 𝑤 ∈ ((𝑘 ∘ ^◡ℎ)‘𝑝))
53	52	ralrimiva 2508	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → ∀𝑝 ∈ ω ∃𝑤 𝑤 ∈ ((𝑘 ∘ ^◡ℎ)‘𝑝))
54	10, 25, 53	cc2 7099	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → ∃𝑔(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚)))
55		simprl 521	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) → 𝑔 Fn ω)
56		f1of 5375	. . . . . . . . . 10 ⊢ (ℎ:𝑁–1-1-onto→ω → ℎ:𝑁⟶ω)
57	56	adantl 275	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → ℎ:𝑁⟶ω)
58	57	adantr 274	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) → ℎ:𝑁⟶ω)
59		fnfco 5305	. . . . . . . 8 ⊢ ((𝑔 Fn ω ∧ ℎ:𝑁⟶ω) → (𝑔 ∘ ℎ) Fn 𝑁)
60	55, 58, 59	syl2anc 409	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) → (𝑔 ∘ ℎ) Fn 𝑁)
61		nfv 1509	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜑
62		nfmpt1 4029	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑁 ↦ 𝐹)
63	62	nfeq2 2294	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)
64	61, 63	nfan 1545	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹))
65		nfv 1509	. . . . . . . . . 10 ⊢ Ⅎ𝑛 ℎ:𝑁–1-1-onto→ω
66	64, 65	nfan 1545	. . . . . . . . 9 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω)
67		nfv 1509	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))
68	66, 67	nfan 1545	. . . . . . . 8 ⊢ Ⅎ𝑛(((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚)))
69		fvco3 5500	. . . . . . . . . . . . . 14 ⊢ ((ℎ:𝑁⟶ω ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) = (𝑔‘(ℎ‘𝑛)))
70	58, 69	sylan 281	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) = (𝑔‘(ℎ‘𝑛)))
71		fveq2 5429	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (ℎ‘𝑛) → (𝑔‘𝑚) = (𝑔‘(ℎ‘𝑛)))
72		fveq2 5429	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (ℎ‘𝑛) → ((𝑘 ∘ ^◡ℎ)‘𝑚) = ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)))
73	71, 72	eleq12d 2211	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (ℎ‘𝑛) → ((𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚) ↔ (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛))))
74		simplrr 526	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))
75	58	ffvelrnda 5563	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → (ℎ‘𝑛) ∈ ω)
76	73, 74, 75	rspcdva 2798	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)))
77	70, 76	eqeltrd 2217	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) ∈ ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)))
78	23	ad2antrr 480	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ^◡ℎ:ω⟶𝑁)
79		fvco3 5500	. . . . . . . . . . . . 13 ⊢ ((^◡ℎ:ω⟶𝑁 ∧ (ℎ‘𝑛) ∈ ω) → ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) = (𝑘‘(^◡ℎ‘(ℎ‘𝑛))))
80	78, 75, 79	syl2anc 409	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) = (𝑘‘(^◡ℎ‘(ℎ‘𝑛))))
81	77, 80	eleqtrd 2219	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) ∈ (𝑘‘(^◡ℎ‘(ℎ‘𝑛))))
82		simpllr 524	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ℎ:𝑁–1-1-onto→ω)
83		simpr 109	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → 𝑛 ∈ 𝑁)
84		f1ocnvfv1 5686	. . . . . . . . . . . . 13 ⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (^◡ℎ‘(ℎ‘𝑛)) = 𝑛)
85	82, 83, 84	syl2anc 409	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → (^◡ℎ‘(ℎ‘𝑛)) = 𝑛)
86	85	fveq2d 5433	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → (𝑘‘(^◡ℎ‘(ℎ‘𝑛))) = (𝑘‘𝑛))
87	81, 86	eleqtrd 2219	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) ∈ (𝑘‘𝑛))
88	16	ad3antrrr 484	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹))
89	88	fveq1d 5431	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → (𝑘‘𝑛) = ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛))
90	11	r19.21bi 2523	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐹 ∈ V)
91	90	ad5ant15 513	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → 𝐹 ∈ V)
92	12	fvmpt2 5512	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ 𝑁 ∧ 𝐹 ∈ V) → ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛) = 𝐹)
93	83, 91, 92	syl2anc 409	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛) = 𝐹)
94	89, 93	eqtrd 2173	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → (𝑘‘𝑛) = 𝐹)
95	87, 94	eleqtrd 2219	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)
96	95	ex 114	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) → (𝑛 ∈ 𝑁 → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))
97	68, 96	ralrimi 2506	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) → ∀𝑛 ∈ 𝑁 ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)
98		vex 2692	. . . . . . . . 9 ⊢ 𝑔 ∈ V
99		vex 2692	. . . . . . . . 9 ⊢ ℎ ∈ V
100	98, 99	coex 5092	. . . . . . . 8 ⊢ (𝑔 ∘ ℎ) ∈ V
101		fneq1 5219	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓 Fn 𝑁 ↔ (𝑔 ∘ ℎ) Fn 𝑁))
102		fveq1 5428	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓‘𝑛) = ((𝑔 ∘ ℎ)‘𝑛))
103	102	eleq1d 2209	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑛) ∈ 𝐹 ↔ ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))
104	103	ralbidv 2438	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹 ↔ ∀𝑛 ∈ 𝑁 ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))
105	101, 104	anbi12d 465	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹) ↔ ((𝑔 ∘ ℎ) Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)))
106	100, 105	spcev 2784	. . . . . . 7 ⊢ (((𝑔 ∘ ℎ) Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹))
107	60, 97, 106	syl2anc 409	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹))
108	54, 107	exlimddv 1871	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹))
109	8, 108	exlimddv 1871	. . . 4 ⊢ ((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹))
110	109	expcom 115	. . 3 ⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹)))
111	110	vtocleg 2760	. 2 ⊢ ((𝑛 ∈ 𝑁 ↦ 𝐹) ∈ V → (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹)))
112	5, 111	mpcom 36	1 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∃wex 1469 ∈ wcel 1481 ∀wral 2417 Vcvv 2689 ⦋csb 3007 class class class wbr 3937 ↦ cmpt 3997 ωcom 4512 ^◡ccnv 4546 ∘ ccom 4551 Fn wfn 5126 ⟶wf 5127 –1-1-onto→wf1o 5130 ‘cfv 5131 ≈ cen 6640 CCHOICEwacc 7094

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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-2nd 6047 df-er 6437 df-en 6643 df-cc 7095

This theorem is referenced by: cc4f 7101 cc4n 7103

W3C validator