Theorem fucpropd 18054

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 17976	. . . 4 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
10	9	opeq2d 4847	. . 3 ⊢ (𝜑 → ⟨(Base‘ndx), (𝐴 Func 𝐶)⟩ = ⟨(Base‘ndx), (𝐵 Func 𝐷)⟩)
11	1, 2, 3, 4, 5, 6, 7, 8	natpropd 18053	. . . 4 ⊢ (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
12	11	opeq2d 4847	. . 3 ⊢ (𝜑 → ⟨(Hom ‘ndx), (𝐴 Nat 𝐶)⟩ = ⟨(Hom ‘ndx), (𝐵 Nat 𝐷)⟩)
13	9	sqxpeqd 5695	. . . . 5 ⊢ (𝜑 → ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) = ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)))
14	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶))) → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
15		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑓(𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶)))
16		nfcsb1v 3878	. . . . . . 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 6900	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) → (1st ‘𝑣) ∈ V)
19		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑔((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣))
20		nfcsb1v 3878	. . . . . . . . 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 6900	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ℎ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st ‘𝑣)) → (2nd ‘𝑣) ∈ V)
23	11	ad3antrrr 743	. . . . . . . . . . 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 7433	. . . . . . . . . 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 7444	. . . . . . . . . 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 17769	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
27	26	ad4antr 745	. . . . . . . . . . 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 2765	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
29		eqid 2765	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
30		eqid 2765	. . . . . . . . . . . 12 ⊢ (comp‘𝐶) = (comp‘𝐶)
31		eqid 2765	. . . . . . . . . . . 12 ⊢ (comp‘𝐷) = (comp‘𝐷)
32	3	ad5antr 747	. . . . . . . . . . . 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 747	. . . . . . . . . . . 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 2765	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐴) = (Base‘𝐴)
35		relfunc 17936	. . . . . . . . . . . . . . 15 ⊢ Rel (𝐴 Func 𝐶)
36		simpllr 788	. . . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . . . 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 8020	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) → (1st ‘𝑣) ∈ (𝐴 Func 𝐶))
40	38, 39	syl 18	. . . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . . 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 8041	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝐴 Func 𝐶) ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (1st ‘𝑓)(𝐴 Func 𝐶)(2nd ‘𝑓))
43	35, 41, 42	sylancr 599	. . . . . . . . . . . . . 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 17940	. . . . . . . . . . . . 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	ffvelcdmda 7083	. . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . 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 8021	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) → (2nd ‘𝑣) ∈ (𝐴 Func 𝐶))
48	38, 47	syl 18	. . . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . . 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 8041	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)) → (1st ‘𝑔)(𝐴 Func 𝐶)(2nd ‘𝑔))
51	35, 49, 50	sylancr 599	. . . . . . . . . . . . . 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 17940	. . . . . . . . . . . . 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	ffvelcdmda 7083	. . . . . . . . . . . 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 501	. . . . . . . . . . . . . . 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 8041	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝐴 Func 𝐶) ∧ ℎ ∈ (𝐴 Func 𝐶)) → (1st ‘ℎ)(𝐴 Func 𝐶)(2nd ‘ℎ))
56	35, 54, 55	sylancr 599	. . . . . . . . . . . . . 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 17940	. . . . . . . . . . . . 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	ffvelcdmda 7083	. . . . . . . . . . . 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 2765	. . . . . . . . . . . . 13 ⊢ (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
60		simplrr 790	. . . . . . . . . . . . . 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 18028	. . . . . . . . . . . . 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 490	. . . . . . . . . . . . 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 18030	. . . . . . . . . . . 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 789	. . . . . . . . . . . . . 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 18028	. . . . . . . . . . . . 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 18030	. . . . . . . . . . . 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 17781	. . . . . . . . . . 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 5199	. . . . . . . . . 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 7490	. . . . . . . . 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 3868	. . . . . . . . . 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 487	. . . . . . . . 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 2800	. . . . . . . 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 3884	. . . . . . 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 3868	. . . . . . . 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 487	. . . . . . 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 2800	. . . . . 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 3884	. . . . 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 7490	. . . 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 4847	. . 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 4716	. 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 2765	. . 3 ⊢ (𝐴 FuncCat 𝐶) = (𝐴 FuncCat 𝐶)
82		eqid 2765	. . 3 ⊢ (𝐴 Func 𝐶) = (𝐴 Func 𝐶)
83		eqidd 2766	. . 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 18035	. 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 2765	. . 3 ⊢ (𝐵 FuncCat 𝐷) = (𝐵 FuncCat 𝐷)
86		eqid 2765	. . 3 ⊢ (𝐵 Func 𝐷) = (𝐵 Func 𝐷)
87		eqid 2765	. . 3 ⊢ (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
88		eqid 2765	. . 3 ⊢ (Base‘𝐵) = (Base‘𝐵)
89		eqidd 2766	. . 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 18035	. 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 2810	1 ⊢ (𝜑 → (𝐴 FuncCat 𝐶) = (𝐵 FuncCat 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Ⅎwnfc 2912  Vcvv 3457  ⦋csb 3854  {ctp 4595  ⟨cop 4597   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661  Rel wrel 5668  ‘cfv 6540  (class class class)co 7416   ∈ cmpo 7418  1^st c1st 7986  2^nd c2nd 7987  ndxcnx 17270  Basecbs 17286  Hom chom 17338  compcco 17339  Catccat 17737  Hom_f chomf 17739  comp_fccomf 17740   Func cfunc 17928   Nat cnat 18018   FuncCat cfuc 18019

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 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-map 8828  df-ixp 8898  df-cat 17741  df-cid 17742  df-homf 17743  df-comf 17744  df-func 17932  df-nat 18020  df-fuc 18021

This theorem is used by:  oyoncl  18343  lanpropd  50426  ranpropd  50427  lmdpropd  50468  cmdpropd  50469  cmddu  50479