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Theorem axcc3 10432

Description: A possibly more useful version of ax-cc 10429 using sequences 𝐹(𝑛) instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)

**Hypotheses**
Ref	Expression
axcc3.1	⊢ 𝐹 ∈ V
axcc3.2	⊢ 𝑁 ≈ ω

**Assertion**
Ref	Expression
axcc3	⊢ ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹))

Distinct variable groups: 𝑓,𝐹 𝑓,𝑁,𝑛 Allowed substitution hint: 𝐹(𝑛)

Proof of Theorem axcc3 Dummy variables 𝑔 ℎ 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		axcc3.2	. . 3 ⊢ 𝑁 ≈ ω
2		relen 8950	. . . 4 ⊢ Rel ≈
3	2	brrelex1i 5720	. . 3 ⊢ (𝑁 ≈ ω → 𝑁 ∈ V)
4		mptexg 7223	. . 3 ⊢ (𝑁 ∈ V → (𝑛 ∈ 𝑁 ↦ 𝐹) ∈ V)
5	1, 3, 4	mp2b 10	. 2 ⊢ (𝑛 ∈ 𝑁 ↦ 𝐹) ∈ V
6		bren 8955	. . . 4 ⊢ (𝑁 ≈ ω ↔ ∃ℎ ℎ:𝑁–1-1-onto→ω)
7	1, 6	mpbi 233	. . 3 ⊢ ∃ℎ ℎ:𝑁–1-1-onto→ω
8		axcc2 10431	. . . . 5 ⊢ ∃𝑔(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚)))
9		f1of 6824	. . . . . . . . . . 11 ⊢ (ℎ:𝑁–1-1-onto→ω → ℎ:𝑁⟶ω)
10		fnfco 6747	. . . . . . . . . . 11 ⊢ ((𝑔 Fn ω ∧ ℎ:𝑁⟶ω) → (𝑔 ∘ ℎ) Fn 𝑁)
11	9, 10	sylan2 605	. . . . . . . . . 10 ⊢ ((𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) → (𝑔 ∘ ℎ) Fn 𝑁)
12	11	adantlr 728	. . . . . . . . 9 ⊢ (((𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ ℎ:𝑁–1-1-onto→ω) → (𝑔 ∘ ℎ) Fn 𝑁)
13	12	3adant1 1148	. . . . . . . 8 ⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ ℎ:𝑁–1-1-onto→ω) → (𝑔 ∘ ℎ) Fn 𝑁)
14		nfmpt1 5213	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑁 ↦ 𝐹)
15	14	nfeq2 2945	. . . . . . . . . 10 ⊢ Ⅎ𝑛 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)
16		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚)))
17		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑛 ℎ:𝑁–1-1-onto→ω
18	15, 16, 17	nf3an 1934	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ ℎ:𝑁–1-1-onto→ω)
19	9	ffvelcdmda 7083	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (ℎ‘𝑛) ∈ ω)
20		fveq2 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = (ℎ‘𝑛) → ((𝑘 ∘ ^◡ℎ)‘𝑚) = ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)))
21	20	neeq1d 3020	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = (ℎ‘𝑛) → (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ ↔ ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) ≠ ∅))
22		fveq2 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = (ℎ‘𝑛) → (𝑔‘𝑚) = (𝑔‘(ℎ‘𝑛)))
23	22, 20	eleq12d 2860	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = (ℎ‘𝑛) → ((𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚) ↔ (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛))))
24	21, 23	imbi12d 347	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = (ℎ‘𝑛) → ((((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚)) ↔ (((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) ≠ ∅ → (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)))))
25	24	rspcv 3580	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℎ‘𝑛) ∈ ω → (∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚)) → (((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) ≠ ∅ → (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)))))
26	19, 25	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚)) → (((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) ≠ ∅ → (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)))))
27	26	3ad2antl3 1206	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → (∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚)) → (((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) ≠ ∅ → (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)))))
28		f1ocnv 6837	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ:𝑁–1-1-onto→ω → ^◡ℎ:ω–1-1-onto→𝑁)
29		f1of 6824	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (^◡ℎ:ω–1-1-onto→𝑁 → ^◡ℎ:ω⟶𝑁)
30	28, 29	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ:𝑁–1-1-onto→ω → ^◡ℎ:ω⟶𝑁)
31		fvco3 6985	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((^◡ℎ:ω⟶𝑁 ∧ (ℎ‘𝑛) ∈ ω) → ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) = (𝑘‘(^◡ℎ‘(ℎ‘𝑛))))
32	30, 19, 31	syl2an2r 698	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) = (𝑘‘(^◡ℎ‘(ℎ‘𝑛))))
33	32	3adant1 1148	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) = (𝑘‘(^◡ℎ‘(ℎ‘𝑛))))
34		f1ocnvfv1 7278	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (^◡ℎ‘(ℎ‘𝑛)) = 𝑛)
35	34	fveq2d 6889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (𝑘‘(^◡ℎ‘(ℎ‘𝑛))) = (𝑘‘𝑛))
36	35	3adant1 1148	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (𝑘‘(^◡ℎ‘(ℎ‘𝑛))) = (𝑘‘𝑛))
37		fveq1 6884	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (𝑘‘𝑛) = ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛))
38		axcc3.1	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝐹 ∈ V
39		eqid 2766	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ 𝑁 ↦ 𝐹) = (𝑛 ∈ 𝑁 ↦ 𝐹)
40	39	fvmpt2 7005	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 ∈ 𝑁 ∧ 𝐹 ∈ V) → ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛) = 𝐹)
41	38, 40	mpan2 704	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ 𝑁 → ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛) = 𝐹)
42	37, 41	sylan9eq 2821	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑛 ∈ 𝑁) → (𝑘‘𝑛) = 𝐹)
43	42	3adant2 1149	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (𝑘‘𝑛) = 𝐹)
44	33, 36, 43	3eqtrd 2805	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) = 𝐹)
45	44	3expa 1136	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) = 𝐹)
46	45	3adantl2 1186	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) = 𝐹)
47	46	neeq1d 3020	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → (((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) ≠ ∅ ↔ 𝐹 ≠ ∅))
48	9	3ad2ant3 1153	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) → ℎ:𝑁⟶ω)
49		fvco3 6985	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ:𝑁⟶ω ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) = (𝑔‘(ℎ‘𝑛)))
50	48, 49	sylan 592	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) = (𝑔‘(ℎ‘𝑛)))
51	50	eleq1d 2851	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → (((𝑔 ∘ ℎ)‘𝑛) ∈ ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) ↔ (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛))))
52	46	eleq2d 2852	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → (((𝑔 ∘ ℎ)‘𝑛) ∈ ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) ↔ ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))
53	51, 52	bitr3d 284	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → ((𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) ↔ ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))
54	47, 53	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → ((((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛)) ≠ ∅ → (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ^◡ℎ)‘(ℎ‘𝑛))) ↔ (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)))
55	27, 54	sylibd 242	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → (∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚)) → (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)))
56	55	ex 418	. . . . . . . . . . . . . 14 ⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) → (𝑛 ∈ 𝑁 → (∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚)) → (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))))
57	56	com23 87	. . . . . . . . . . . . 13 ⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) → (∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚)) → (𝑛 ∈ 𝑁 → (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))))
58	57	3exp 1137	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (𝑔 Fn ω → (ℎ:𝑁–1-1-onto→ω → (∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚)) → (𝑛 ∈ 𝑁 → (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))))))
59	58	com34 92	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (𝑔 Fn ω → (∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚)) → (ℎ:𝑁–1-1-onto→ω → (𝑛 ∈ 𝑁 → (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))))))
60	59	imp32 424	. . . . . . . . . 10 ⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚)))) → (ℎ:𝑁–1-1-onto→ω → (𝑛 ∈ 𝑁 → (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))))
61	60	3impia 1135	. . . . . . . . 9 ⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ ℎ:𝑁–1-1-onto→ω) → (𝑛 ∈ 𝑁 → (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)))
62	18, 61	ralrimi 3266	. . . . . . . 8 ⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ ℎ:𝑁–1-1-onto→ω) → ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))
63		vex 3462	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
64		vex 3462	. . . . . . . . . 10 ⊢ ℎ ∈ V
65	63, 64	coex 7929	. . . . . . . . 9 ⊢ (𝑔 ∘ ℎ) ∈ V
66		fneq1 6630	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓 Fn 𝑁 ↔ (𝑔 ∘ ℎ) Fn 𝑁))
67		fveq1 6884	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓‘𝑛) = ((𝑔 ∘ ℎ)‘𝑛))
68	67	eleq1d 2851	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑛) ∈ 𝐹 ↔ ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))
69	68	imbi2d 343	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹) ↔ (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)))
70	69	ralbidv 3191	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹) ↔ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)))
71	66, 70	anbi12d 644	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹)) ↔ ((𝑔 ∘ ℎ) Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))))
72	65, 71	spcev 3568	. . . . . . . 8 ⊢ (((𝑔 ∘ ℎ) Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹)))
73	13, 62, 72	syl2anc 596	. . . . . . 7 ⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) ∧ ℎ:𝑁–1-1-onto→ω) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹)))
74	73	3exp 1137	. . . . . 6 ⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → ((𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) → (ℎ:𝑁–1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹)))))
75	74	exlimdv 1966	. . . . 5 ⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (∃𝑔(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ^◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ^◡ℎ)‘𝑚))) → (ℎ:𝑁–1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹)))))
76	8, 75	mpi 21	. . . 4 ⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (ℎ:𝑁–1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹))))
77	76	exlimdv 1966	. . 3 ⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (∃ℎ ℎ:𝑁–1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹))))
78	7, 77	mpi 21	. 2 ⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹)))
79	5, 78	vtocle 3526	1 ⊢ ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2146 ≠ wne 2961 ∀wral 3082 Vcvv 3458 ∅c0 4289 class class class wbr 5112 ↦ cmpt 5195 ^◡ccnv 5663 ∘ ccom 5668 Fn wfn 6535 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 ωcom 7864 ≈ cen 8942

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5241 ax-sep 5260 ax-nul 5272 ax-pow 5339 ax-pr 5407 ax-un 7738 ax-inf2 9612 ax-cc 10429

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4876 df-iun 4961 df-br 5113 df-opab 5177 df-mpt 5196 df-tr 5222 df-id 5559 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-om 7865 df-2nd 7989 df-er 8696 df-en 8946

This theorem is used by: axcc4 10433 domtriomlem 10436 ovnsubaddlem2 47317

W3C validator