Theorem natpropd 18160

Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same natural transformations. (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
natpropd	⊢ (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))

Proof of Theorem natpropd

Dummy variables 𝑎 𝑓 𝑔 ℎ 𝑟 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucpropd.1	. . . 4 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
2		fucpropd.2	. . . 4 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
3		fucpropd.3	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
4		fucpropd.4	. . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
5		fucpropd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Cat)
6		fucpropd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ Cat)
7		fucpropd.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
8		fucpropd.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
9	1, 2, 3, 4, 5, 6, 7, 8	funcpropd 18083	. . 3 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
10	9	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
11		nfv 1946	. . . 4 ⊢ Ⅎ𝑟(𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)))
12		nfcsb1v 3870	. . . . 5 ⊢ Ⅎ𝑟⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))}
13	12	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) → Ⅎ𝑟⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
14		fvexd 6901	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) → (1st ‘𝑓) ∈ V)
15		nfv 1946	. . . . . 6 ⊢ Ⅎ𝑠((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓))
16		nfcsb1v 3870	. . . . . . 7 ⊢ Ⅎ𝑠⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))}
17	16	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) → Ⅎ𝑠⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
18		fvexd 6901	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) → (1st ‘𝑔) ∈ V)
19		eqid 2760	. . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶)
20		eqid 2760	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
21		eqid 2760	. . . . . . . . . . 11 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
22	3	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → (Homf ‘𝐶) = (Homf ‘𝐷))
23		eqid 2760	. . . . . . . . . . . . 13 ⊢ (Base‘𝐴) = (Base‘𝐴)
24		simplr 781	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → 𝑟 = (1st ‘𝑓))
25		relfunc 18043	. . . . . . . . . . . . . . 15 ⊢ Rel (𝐴 Func 𝐶)
26		simpllr 788	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)))
27	26	simpld 500	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → 𝑓 ∈ (𝐴 Func 𝐶))
28		1st2ndbr 8054	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝐴 Func 𝐶) ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (1st ‘𝑓)(𝐴 Func 𝐶)(2nd ‘𝑓))
29	25, 27, 28	sylancr 599	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → (1st ‘𝑓)(𝐴 Func 𝐶)(2nd ‘𝑓))
30	24, 29	eqbrtrd 5126	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → 𝑟(𝐴 Func 𝐶)(2nd ‘𝑓))
31	23, 19, 30	funcf1 18047	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → 𝑟:(Base‘𝐴)⟶(Base‘𝐶))
32	31	ffvelcdmda 7085	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑟‘𝑥) ∈ (Base‘𝐶))
33		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → 𝑠 = (1st ‘𝑔))
34	26	simprd 501	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → 𝑔 ∈ (𝐴 Func 𝐶))
35		1st2ndbr 8054	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)) → (1st ‘𝑔)(𝐴 Func 𝐶)(2nd ‘𝑔))
36	25, 34, 35	sylancr 599	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → (1st ‘𝑔)(𝐴 Func 𝐶)(2nd ‘𝑔))
37	33, 36	eqbrtrd 5126	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → 𝑠(𝐴 Func 𝐶)(2nd ‘𝑔))
38	23, 19, 37	funcf1 18047	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → 𝑠:(Base‘𝐴)⟶(Base‘𝐶))
39	38	ffvelcdmda 7085	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑠‘𝑥) ∈ (Base‘𝐶))
40	19, 20, 21, 22, 32, 39	homfeqval 17877	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → ((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)) = ((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)))
41	40	ixpeq2dva 8944	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → X𝑥 ∈ (Base‘𝐴)((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)) = X𝑥 ∈ (Base‘𝐴)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)))
42	1	homfeqbas 17876	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
43	42	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → (Base‘𝐴) = (Base‘𝐵))
44	43	ixpeq1d 8941	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → X𝑥 ∈ (Base‘𝐴)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) = X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)))
45	41, 44	eqtrd 2795	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → X𝑥 ∈ (Base‘𝐴)((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)) = X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)))
46		fveq2 6886	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑟‘𝑥) = (𝑟‘𝑧))
47		fveq2 6886	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑠‘𝑥) = (𝑠‘𝑧))
48	46, 47	oveq12d 7439	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)) = ((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧)))
49	48	cbvixpv 8947	. . . . . . . . . 10 ⊢ X𝑥 ∈ (Base‘𝐴)((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)) = X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))
50	49	eleq2i 2852	. . . . . . . . 9 ⊢ (𝑎 ∈ X𝑥 ∈ (Base‘𝐴)((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)) ↔ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧)))
51	43	adantr 486	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) → (Base‘𝐴) = (Base‘𝐵))
52	51	adantr 486	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) → (Base‘𝐴) = (Base‘𝐵))
53		eqid 2760	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
54		eqid 2760	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
55	1	ad6antr 749	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (Homf ‘𝐴) = (Homf ‘𝐵))
56		simplr 781	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑥 ∈ (Base‘𝐴))
57		simpr 490	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑦 ∈ (Base‘𝐴))
58	23, 53, 54, 55, 56, 57	homfeqval 17877	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
59		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (comp‘𝐶) = (comp‘𝐶)
60		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (comp‘𝐷) = (comp‘𝐷)
61	3	ad7antr 751	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (Homf ‘𝐶) = (Homf ‘𝐷))
62	4	ad7antr 751	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (compf‘𝐶) = (compf‘𝐷))
63	32	ad5ant13 769	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑟‘𝑥) ∈ (Base‘𝐶))
64	31	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑟:(Base‘𝐴)⟶(Base‘𝐶))
65	64	ffvelcdmda 7085	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑟‘𝑦) ∈ (Base‘𝐶))
66	65	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑟‘𝑦) ∈ (Base‘𝐶))
67	38	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑠:(Base‘𝐴)⟶(Base‘𝐶))
68	67	ffvelcdmda 7085	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑠‘𝑦) ∈ (Base‘𝐶))
69	68	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑠‘𝑦) ∈ (Base‘𝐶))
70	30	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑟(𝐴 Func 𝐶)(2nd ‘𝑓))
71	23, 53, 20, 70, 56, 57	funcf2 18049	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑥(2nd ‘𝑓)𝑦):(𝑥(Hom ‘𝐴)𝑦)⟶((𝑟‘𝑥)(Hom ‘𝐶)(𝑟‘𝑦)))
72	71	ffvelcdmda 7085	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)) → ((𝑥(2nd ‘𝑓)𝑦)‘ℎ) ∈ ((𝑟‘𝑥)(Hom ‘𝐶)(𝑟‘𝑦)))
73		fveq2 6886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑦 → (𝑟‘𝑧) = (𝑟‘𝑦))
74		fveq2 6886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑦 → (𝑠‘𝑧) = (𝑠‘𝑦))
75	73, 74	oveq12d 7439	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 → ((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧)) = ((𝑟‘𝑦)(Hom ‘𝐶)(𝑠‘𝑦)))
76	75	fvixp 8934	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑎‘𝑦) ∈ ((𝑟‘𝑦)(Hom ‘𝐶)(𝑠‘𝑦)))
77	76	ad5ant24 773	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑎‘𝑦) ∈ ((𝑟‘𝑦)(Hom ‘𝐶)(𝑠‘𝑦)))
78	19, 20, 59, 60, 61, 62, 63, 66, 69, 72, 77	comfeqval 17888	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)) → ((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐶)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = ((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)))
79	39	ad5ant13 769	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑠‘𝑥) ∈ (Base‘𝐶))
80		fveq2 6886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑥 → (𝑟‘𝑧) = (𝑟‘𝑥))
81		fveq2 6886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑥 → (𝑠‘𝑧) = (𝑠‘𝑥))
82	80, 81	oveq12d 7439	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → ((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧)) = ((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)))
83	82	fvixp 8934	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧)) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑎‘𝑥) ∈ ((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)))
84	83	ad5ant23 772	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑎‘𝑥) ∈ ((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)))
85	37	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑠(𝐴 Func 𝐶)(2nd ‘𝑔))
86	23, 53, 20, 85, 56, 57	funcf2 18049	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑥(2nd ‘𝑔)𝑦):(𝑥(Hom ‘𝐴)𝑦)⟶((𝑠‘𝑥)(Hom ‘𝐶)(𝑠‘𝑦)))
87	86	ffvelcdmda 7085	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)) → ((𝑥(2nd ‘𝑔)𝑦)‘ℎ) ∈ ((𝑠‘𝑥)(Hom ‘𝐶)(𝑠‘𝑦)))
88	19, 20, 59, 60, 61, 62, 63, 79, 69, 84, 87	comfeqval 17888	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐶)(𝑠‘𝑦))(𝑎‘𝑥)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥)))
89	78, 88	eqeq12d 2776	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐶)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐶)(𝑠‘𝑦))(𝑎‘𝑥)) ↔ ((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))))
90	58, 89	raleqbidva 3325	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (∀ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐶)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐶)(𝑠‘𝑦))(𝑎‘𝑥)) ↔ ∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))))
91	52, 90	raleqbidva 3325	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) → (∀𝑦 ∈ (Base‘𝐴)∀ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐶)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐶)(𝑠‘𝑦))(𝑎‘𝑥)) ↔ ∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))))
92	51, 91	raleqbidva 3325	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑧 ∈ (Base‘𝐴)((𝑟‘𝑧)(Hom ‘𝐶)(𝑠‘𝑧))) → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐶)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐶)(𝑠‘𝑦))(𝑎‘𝑥)) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))))
93	50, 92	sylan2b 606	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) ∧ 𝑎 ∈ X𝑥 ∈ (Base‘𝐴)((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥))) → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐶)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐶)(𝑠‘𝑦))(𝑎‘𝑥)) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))))
94	45, 93	rabeqbidva 3428	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → {𝑎 ∈ X𝑥 ∈ (Base‘𝐴)((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐶)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐶)(𝑠‘𝑦))(𝑎‘𝑥))} = {𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
95		csbeq1a 3860	. . . . . . . 8 ⊢ (𝑠 = (1st ‘𝑔) → {𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} = ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
96	95	adantl 487	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → {𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} = ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
97	94, 96	eqtrd 2795	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) ∧ 𝑠 = (1st ‘𝑔)) → {𝑎 ∈ X𝑥 ∈ (Base‘𝐴)((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐶)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐶)(𝑠‘𝑦))(𝑎‘𝑥))} = ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
98	15, 17, 18, 97	csbiedf 3876	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) → ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐴)((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐶)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐶)(𝑠‘𝑦))(𝑎‘𝑥))} = ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
99		csbeq1a 3860	. . . . . 6 ⊢ (𝑟 = (1st ‘𝑓) → ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} = ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
100	99	adantl 487	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) → ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} = ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
101	98, 100	eqtrd 2795	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st ‘𝑓)) → ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐴)((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐶)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐶)(𝑠‘𝑦))(𝑎‘𝑥))} = ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
102	11, 13, 14, 101	csbiedf 3876	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) → ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐴)((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐶)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐶)(𝑠‘𝑦))(𝑎‘𝑥))} = ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
103	9, 10, 102	mpoeq123dva 7495	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐴 Func 𝐶), 𝑔 ∈ (𝐴 Func 𝐶) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐴)((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐶)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐶)(𝑠‘𝑦))(𝑎‘𝑥))}) = (𝑓 ∈ (𝐵 Func 𝐷), 𝑔 ∈ (𝐵 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))}))
104		eqid 2760	. . 3 ⊢ (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
105	104, 23, 53, 20, 59	natfval 18130	. 2 ⊢ (𝐴 Nat 𝐶) = (𝑓 ∈ (𝐴 Func 𝐶), 𝑔 ∈ (𝐴 Func 𝐶) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐴)((𝑟‘𝑥)(Hom ‘𝐶)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ℎ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐶)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐶)(𝑠‘𝑦))(𝑎‘𝑥))})
106		eqid 2760	. . 3 ⊢ (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
107		eqid 2760	. . 3 ⊢ (Base‘𝐵) = (Base‘𝐵)
108	106, 107, 54, 21, 60	natfval 18130	. 2 ⊢ (𝐵 Nat 𝐷) = (𝑓 ∈ (𝐵 Func 𝐷), 𝑔 ∈ (𝐵 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐵)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ℎ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
109	103, 105, 108	3eqtr4g 2820	1 ⊢ (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Ⅎwnfc 2907  ∀wral 3076  {crab 3412  Vcvv 3450  ⦋csb 3846  ⟨cop 4834   class class class wbr 5102  Rel wrel 5655  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423  1^st c1st 8000  2^nd c2nd 8001  Xcixp 8929  Basecbs 17393  Hom chom 17445  compcco 17446  Catccat 17844  Hom_f chomf 17846  comp_fccomf 17847   Func cfunc 18035   Nat cnat 18125

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-map 8853  df-ixp 8930  df-cat 17848  df-cid 17849  df-homf 17850  df-comf 17851  df-func 18039  df-nat 18127

This theorem is used by:  fucpropd  18161  natoppfb  50447  prcofpropd  50608