Theorem fucpropd 17230

Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same functor categories. (Contributed by Mario Carneiro, 26-Jan-2017.)

Hypotheses

Ref	Expression
fucpropd.1	⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
fucpropd.2	⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
fucpropd.3	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
fucpropd.4	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
fucpropd.a	⊢ (𝜑 → 𝐴 ∈ Cat)
fucpropd.b	⊢ (𝜑 → 𝐵 ∈ Cat)
fucpropd.c	⊢ (𝜑 → 𝐶 ∈ Cat)
fucpropd.d	⊢ (𝜑 → 𝐷 ∈ Cat)

Assertion

Ref	Expression
fucpropd	⊢ (𝜑 → (𝐴 FuncCat 𝐶) = (𝐵 FuncCat 𝐷))

Proof of Theorem fucpropd

Dummy variables 𝑎 𝑏 𝑓 𝑔 ℎ 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucpropd.1	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
2		fucpropd.2	. . . . 5 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
3		fucpropd.3	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
4		fucpropd.4	. . . . 5 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
5		fucpropd.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Cat)
6		fucpropd.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ Cat)
7		fucpropd.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
8		fucpropd.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
9	1, 2, 3, 4, 5, 6, 7, 8	funcpropd 17153	. . . 4 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
10	9	opeq2d 4796	. . 3 ⊢ (𝜑 → ⟨(Base‘ndx), (𝐴 Func 𝐶)⟩ = ⟨(Base‘ndx), (𝐵 Func 𝐷)⟩)
11	1, 2, 3, 4, 5, 6, 7, 8	natpropd 17229	. . . 4 ⊢ (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
12	11	opeq2d 4796	. . 3 ⊢ (𝜑 → ⟨(Hom ‘ndx), (𝐴 Nat 𝐶)⟩ = ⟨(Hom ‘ndx), (𝐵 Nat 𝐷)⟩)
13	9	sqxpeqd 5573	. . . . 5 ⊢ (𝜑 → ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) = ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)))
14	9	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶))) → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
15		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑓(𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶)))
16		nfcsb1v 3895	. . . . . . 7 ⊢ Ⅎ𝑓⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))
17	16	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) → Ⅎ𝑓⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
18		fvexd 6671	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) → (1st ‘𝑣) ∈ V)
19		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑔((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣))
20		nfcsb1v 3895	. . . . . . . . 9 ⊢ Ⅎ𝑔⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))
21	20	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) → Ⅎ𝑔⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
22		fvexd 6671	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) → (2nd ‘𝑣) ∈ V)
23	11	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
24	23	oveqd 7159	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) → (𝑔(𝐴 Nat 𝐶)ℎ) = (𝑔(𝐵 Nat 𝐷)ℎ))
25	23	oveqdr 7170	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ 𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ)) → (𝑓(𝐴 Nat 𝐶)𝑔) = (𝑓(𝐵 Nat 𝐷)𝑔))
26	1	homfeqbas 16949	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
27	26	ad4antr 730	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (Base‘𝐴) = (Base‘𝐵))
28		eqid 2821	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
29		eqid 2821	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
30		eqid 2821	. . . . . . . . . . . 12 ⊢ (comp‘𝐶) = (comp‘𝐶)
31		eqid 2821	. . . . . . . . . . . 12 ⊢ (comp‘𝐷) = (comp‘𝐷)
32	3	ad5antr 732	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → (Homf ‘𝐶) = (Homf ‘𝐷))
33	4	ad5antr 732	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → (compf‘𝐶) = (compf‘𝐷))
34		eqid 2821	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐴) = (Base‘𝐴)
35		relfunc 17115	. . . . . . . . . . . . . . 15 ⊢ Rel (𝐴 Func 𝐶)
36		simpllr 774	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → 𝑓 = (1st ‘𝑣))
37		simp-4r 782	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶)))
38	37	simpld 497	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → 𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)))
39		xp1st 7707	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) → (1st ‘𝑣) ∈ (𝐴 Func 𝐶))
40	38, 39	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (1st ‘𝑣) ∈ (𝐴 Func 𝐶))
41	36, 40	eqeltrd 2913	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → 𝑓 ∈ (𝐴 Func 𝐶))
42		1st2ndbr 7727	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝐴 Func 𝐶) ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (1st ‘𝑓)(𝐴 Func 𝐶)(2nd ‘𝑓))
43	35, 41, 42	sylancr 589	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (1st ‘𝑓)(𝐴 Func 𝐶)(2nd ‘𝑓))
44	34, 28, 43	funcf1 17119	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (1st ‘𝑓):(Base‘𝐴)⟶(Base‘𝐶))
45	44	ffvelrnda 6837	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → ((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐶))
46		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → 𝑔 = (2nd ‘𝑣))
47		xp2nd 7708	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) → (2nd ‘𝑣) ∈ (𝐴 Func 𝐶))
48	38, 47	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (2nd ‘𝑣) ∈ (𝐴 Func 𝐶))
49	46, 48	eqeltrd 2913	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → 𝑔 ∈ (𝐴 Func 𝐶))
50		1st2ndbr 7727	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)) → (1st ‘𝑔)(𝐴 Func 𝐶)(2nd ‘𝑔))
51	35, 49, 50	sylancr 589	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (1st ‘𝑔)(𝐴 Func 𝐶)(2nd ‘𝑔))
52	34, 28, 51	funcf1 17119	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (1st ‘𝑔):(Base‘𝐴)⟶(Base‘𝐶))
53	52	ffvelrnda 6837	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → ((1st ‘𝑔)‘𝑥) ∈ (Base‘𝐶))
54	37	simprd 498	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → ℎ ∈ (𝐴 Func 𝐶))
55		1st2ndbr 7727	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝐴 Func 𝐶) ∧ ℎ ∈ (𝐴 Func 𝐶)) → (1st ‘ℎ)(𝐴 Func 𝐶)(2nd ‘ℎ))
56	35, 54, 55	sylancr 589	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (1st ‘ℎ)(𝐴 Func 𝐶)(2nd ‘ℎ))
57	34, 28, 56	funcf1 17119	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (1st ‘ℎ):(Base‘𝐴)⟶(Base‘𝐶))
58	57	ffvelrnda 6837	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → ((1st ‘ℎ)‘𝑥) ∈ (Base‘𝐶))
59		eqid 2821	. . . . . . . . . . . . 13 ⊢ (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
60		simplrr 776	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))
61	59, 60	nat1st2nd 17204	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑎 ∈ (⟨(1st ‘𝑓), (2nd ‘𝑓)⟩(𝐴 Nat 𝐶)⟨(1st ‘𝑔), (2nd ‘𝑔)⟩))
62		simpr 487	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑥 ∈ (Base‘𝐴))
63	59, 61, 34, 29, 62	natcl 17206	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑎‘𝑥) ∈ (((1st ‘𝑓)‘𝑥)(Hom ‘𝐶)((1st ‘𝑔)‘𝑥)))
64		simplrl 775	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ))
65	59, 64	nat1st2nd 17204	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑏 ∈ (⟨(1st ‘𝑔), (2nd ‘𝑔)⟩(𝐴 Nat 𝐶)⟨(1st ‘ℎ), (2nd ‘ℎ)⟩))
66	59, 65, 34, 29, 62	natcl 17206	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑏‘𝑥) ∈ (((1st ‘𝑔)‘𝑥)(Hom ‘𝐶)((1st ‘ℎ)‘𝑥)))
67	28, 29, 30, 31, 32, 33, 45, 53, 58, 63, 66	comfeqval 16961	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐶)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)) = ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))
68	27, 67	mpteq12dva 5136	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐶)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))) = (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))
69	24, 25, 68	mpoeq123dva 7214	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) → (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐶)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = (𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
70		csbeq1a 3885	. . . . . . . . . 10 ⊢ (𝑔 = (2nd ‘𝑣) → (𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = ⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
71	70	adantl 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) → (𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = ⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
72	69, 71	eqtrd 2856	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) ∧ 𝑔 = (2nd ‘𝑣)) → (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐶)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = ⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
73	19, 21, 22, 72	csbiedf 3901	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) → ⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐶)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = ⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
74		csbeq1a 3885	. . . . . . . 8 ⊢ (𝑓 = (1st ‘𝑣) → ⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
75	74	adantl 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) → ⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
76	73, 75	eqtrd 2856	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) → ⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐶)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
77	15, 17, 18, 76	csbiedf 3901	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) → ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐶)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
78	13, 14, 77	mpoeq123dva 7214	. . . 4 ⊢ (𝜑 → (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)), ℎ ∈ (𝐴 Func 𝐶) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐶)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) = (𝑣 ∈ ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)), ℎ ∈ (𝐵 Func 𝐷) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))
79	78	opeq2d 4796	. . 3 ⊢ (𝜑 → ⟨(comp‘ndx), (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)), ℎ ∈ (𝐴 Func 𝐶) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐶)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩ = ⟨(comp‘ndx), (𝑣 ∈ ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)), ℎ ∈ (𝐵 Func 𝐷) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩)
80	10, 12, 79	tpeq123d 4670	. 2 ⊢ (𝜑 → {⟨(Base‘ndx), (𝐴 Func 𝐶)⟩, ⟨(Hom ‘ndx), (𝐴 Nat 𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)), ℎ ∈ (𝐴 Func 𝐶) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐶)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩} = {⟨(Base‘ndx), (𝐵 Func 𝐷)⟩, ⟨(Hom ‘ndx), (𝐵 Nat 𝐷)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)), ℎ ∈ (𝐵 Func 𝐷) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩})
81		eqid 2821	. . 3 ⊢ (𝐴 FuncCat 𝐶) = (𝐴 FuncCat 𝐶)
82		eqid 2821	. . 3 ⊢ (𝐴 Func 𝐶) = (𝐴 Func 𝐶)
83		eqidd 2822	. . 3 ⊢ (𝜑 → (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)), ℎ ∈ (𝐴 Func 𝐶) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐶)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) = (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)), ℎ ∈ (𝐴 Func 𝐶) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐶)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))
84	81, 82, 59, 34, 30, 5, 7, 83	fucval 17211	. 2 ⊢ (𝜑 → (𝐴 FuncCat 𝐶) = {⟨(Base‘ndx), (𝐴 Func 𝐶)⟩, ⟨(Hom ‘ndx), (𝐴 Nat 𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)), ℎ ∈ (𝐴 Func 𝐶) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)ℎ), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐶)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩})
85		eqid 2821	. . 3 ⊢ (𝐵 FuncCat 𝐷) = (𝐵 FuncCat 𝐷)
86		eqid 2821	. . 3 ⊢ (𝐵 Func 𝐷) = (𝐵 Func 𝐷)
87		eqid 2821	. . 3 ⊢ (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
88		eqid 2821	. . 3 ⊢ (Base‘𝐵) = (Base‘𝐵)
89		eqidd 2822	. . 3 ⊢ (𝜑 → (𝑣 ∈ ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)), ℎ ∈ (𝐵 Func 𝐷) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) = (𝑣 ∈ ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)), ℎ ∈ (𝐵 Func 𝐷) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))
90	85, 86, 87, 88, 31, 6, 8, 89	fucval 17211	. 2 ⊢ (𝜑 → (𝐵 FuncCat 𝐷) = {⟨(Base‘ndx), (𝐵 Func 𝐷)⟩, ⟨(Hom ‘ndx), (𝐵 Nat 𝐷)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)), ℎ ∈ (𝐵 Func 𝐷) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩})
91	80, 84, 90	3eqtr4d 2866	1 ⊢ (𝜑 → (𝐴 FuncCat 𝐶) = (𝐵 FuncCat 𝐷))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Ⅎwnfc 2961  Vcvv 3486  ⦋csb 3871  {ctp 4557  ⟨cop 4559   class class class wbr 5052   ↦ cmpt 5132   × cxp 5539  Rel wrel 5546  ‘cfv 6341  (class class class)co 7142   ∈ cmpo 7144  1^st c1st 7673  2^nd c2nd 7674  ndxcnx 16463  Basecbs 16466  Hom chom 16559  compcco 16560  Catccat 16918  Hom_f chomf 16920  comp_fccomf 16921   Func cfunc 17107   Nat cnat 17194   FuncCat cfuc 17195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5252  ax-pr 5316  ax-un 7447

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3488  df-sbc 3764  df-csb 3872  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-pw 4527  df-sn 4554  df-pr 4556  df-tp 4558  df-op 4560  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5446  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-rn 5552  df-res 5553  df-ima 5554  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-riota 7100  df-ov 7145  df-oprab 7146  df-mpo 7147  df-1st 7675  df-2nd 7676  df-map 8394  df-ixp 8448  df-cat 16922  df-cid 16923  df-homf 16924  df-comf 16925  df-func 17111  df-nat 17196  df-fuc 17197

This theorem is referenced by:  oyoncl  17503