Theorem xpcpropd 18225

Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same product category. (Contributed by Mario Carneiro, 17-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
xpcpropd	⊢ (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))

Proof of Theorem xpcpropd

Dummy variables 𝑓 𝑔 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ (𝐴 ×c 𝐶) = (𝐴 ×c 𝐶)
2		eqid 2736	. . 3 ⊢ (Base‘𝐴) = (Base‘𝐴)
3		eqid 2736	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2736	. . 3 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
5		eqid 2736	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2736	. . 3 ⊢ (comp‘𝐴) = (comp‘𝐴)
7		eqid 2736	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
8		xpcpropd.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
9		xpcpropd.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
10		eqidd 2737	. . 3 ⊢ (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐴) × (Base‘𝐶)))
11	1, 2, 3	xpcbas 18195	. . . . 5 ⊢ ((Base‘𝐴) × (Base‘𝐶)) = (Base‘(𝐴 ×c 𝐶))
12		eqid 2736	. . . . 5 ⊢ (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐴 ×c 𝐶))
13	1, 11, 4, 5, 12	xpchomfval 18196	. . . 4 ⊢ (Hom ‘(𝐴 ×c 𝐶)) = (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1st ‘𝑢)(Hom ‘𝐴)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐶)(2nd ‘𝑣))))
14	13	a1i 11	. . 3 ⊢ (𝜑 → (Hom ‘(𝐴 ×c 𝐶)) = (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1st ‘𝑢)(Hom ‘𝐴)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐶)(2nd ‘𝑣)))))
15		eqidd 2737	. . 3 ⊢ (𝜑 → (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐴)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐶)(2nd ‘𝑦))(2nd ‘𝑓))⟩)) = (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐴)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐶)(2nd ‘𝑦))(2nd ‘𝑓))⟩)))
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15	xpcval 18194	. 2 ⊢ (𝜑 → (𝐴 ×c 𝐶) = {⟨(Base‘ndx), ((Base‘𝐴) × (Base‘𝐶))⟩, ⟨(Hom ‘ndx), (Hom ‘(𝐴 ×c 𝐶))⟩, ⟨(comp‘ndx), (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐴)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐶)(2nd ‘𝑦))(2nd ‘𝑓))⟩))⟩})
17		eqid 2736	. . 3 ⊢ (𝐵 ×c 𝐷) = (𝐵 ×c 𝐷)
18		eqid 2736	. . 3 ⊢ (Base‘𝐵) = (Base‘𝐵)
19		eqid 2736	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
20		eqid 2736	. . 3 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
21		eqid 2736	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
22		eqid 2736	. . 3 ⊢ (comp‘𝐵) = (comp‘𝐵)
23		eqid 2736	. . 3 ⊢ (comp‘𝐷) = (comp‘𝐷)
24		xpcpropd.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
25		xpcpropd.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
26		xpcpropd.1	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
27	26	homfeqbas 17713	. . . 4 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
28		xpcpropd.3	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
29	28	homfeqbas 17713	. . . 4 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
30	27, 29	xpeq12d 5690	. . 3 ⊢ (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐵) × (Base‘𝐷)))
31	26	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Homf ‘𝐴) = (Homf ‘𝐵))
32		xp1st 8025	. . . . . . . 8 ⊢ (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st ‘𝑢) ∈ (Base‘𝐴))
33	32	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1st ‘𝑢) ∈ (Base‘𝐴))
34		xp1st 8025	. . . . . . . 8 ⊢ (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st ‘𝑣) ∈ (Base‘𝐴))
35	34	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1st ‘𝑣) ∈ (Base‘𝐴))
36	2, 4, 20, 31, 33, 35	homfeqval 17714	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → ((1st ‘𝑢)(Hom ‘𝐴)(1st ‘𝑣)) = ((1st ‘𝑢)(Hom ‘𝐵)(1st ‘𝑣)))
37	28	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Homf ‘𝐶) = (Homf ‘𝐷))
38		xp2nd 8026	. . . . . . . 8 ⊢ (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd ‘𝑢) ∈ (Base‘𝐶))
39	38	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2nd ‘𝑢) ∈ (Base‘𝐶))
40		xp2nd 8026	. . . . . . . 8 ⊢ (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd ‘𝑣) ∈ (Base‘𝐶))
41	40	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2nd ‘𝑣) ∈ (Base‘𝐶))
42	3, 5, 21, 37, 39, 41	homfeqval 17714	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → ((2nd ‘𝑢)(Hom ‘𝐶)(2nd ‘𝑣)) = ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣)))
43	36, 42	xpeq12d 5690	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (((1st ‘𝑢)(Hom ‘𝐴)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐶)(2nd ‘𝑣))) = (((1st ‘𝑢)(Hom ‘𝐵)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣))))
44	43	mpoeq3dva 7489	. . . 4 ⊢ (𝜑 → (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1st ‘𝑢)(Hom ‘𝐴)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐶)(2nd ‘𝑣)))) = (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1st ‘𝑢)(Hom ‘𝐵)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣)))))
45	13, 44	eqtrid 2783	. . 3 ⊢ (𝜑 → (Hom ‘(𝐴 ×c 𝐶)) = (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1st ‘𝑢)(Hom ‘𝐵)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣)))))
46	26	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (Homf ‘𝐴) = (Homf ‘𝐵))
47		xpcpropd.2	. . . . . . . . . 10 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
48	47	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (compf‘𝐴) = (compf‘𝐵))
49		simp-4r 783	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))))
50		xp1st 8025	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) → (1st ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)))
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (1st ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)))
52		xp1st 8025	. . . . . . . . . 10 ⊢ ((1st ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st ‘(1st ‘𝑥)) ∈ (Base‘𝐴))
53	51, 52	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (1st ‘(1st ‘𝑥)) ∈ (Base‘𝐴))
54		xp2nd 8026	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) → (2nd ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)))
55	49, 54	syl 17	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (2nd ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)))
56		xp1st 8025	. . . . . . . . . 10 ⊢ ((2nd ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st ‘(2nd ‘𝑥)) ∈ (Base‘𝐴))
57	55, 56	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (1st ‘(2nd ‘𝑥)) ∈ (Base‘𝐴))
58		simpllr 775	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)))
59		xp1st 8025	. . . . . . . . . 10 ⊢ (𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st ‘𝑦) ∈ (Base‘𝐴))
60	58, 59	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (1st ‘𝑦) ∈ (Base‘𝐴))
61		simpr 484	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥))
62		1st2nd2 8032	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
63	49, 62	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
64	63	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) = ((Hom ‘(𝐴 ×c 𝐶))‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
65		df-ov 7413	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))(2nd ‘𝑥)) = ((Hom ‘(𝐴 ×c 𝐶))‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
66	64, 65	eqtr4di 2789	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) = ((1st ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))(2nd ‘𝑥)))
67	1, 11, 4, 5, 12, 51, 55	xpchom 18197	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ((1st ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))(2nd ‘𝑥)) = (((1st ‘(1st ‘𝑥))(Hom ‘𝐴)(1st ‘(2nd ‘𝑥))) × ((2nd ‘(1st ‘𝑥))(Hom ‘𝐶)(2nd ‘(2nd ‘𝑥)))))
68	66, 67	eqtrd 2771	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) = (((1st ‘(1st ‘𝑥))(Hom ‘𝐴)(1st ‘(2nd ‘𝑥))) × ((2nd ‘(1st ‘𝑥))(Hom ‘𝐶)(2nd ‘(2nd ‘𝑥)))))
69	61, 68	eleqtrd 2837	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → 𝑓 ∈ (((1st ‘(1st ‘𝑥))(Hom ‘𝐴)(1st ‘(2nd ‘𝑥))) × ((2nd ‘(1st ‘𝑥))(Hom ‘𝐶)(2nd ‘(2nd ‘𝑥)))))
70		xp1st 8025	. . . . . . . . . 10 ⊢ (𝑓 ∈ (((1st ‘(1st ‘𝑥))(Hom ‘𝐴)(1st ‘(2nd ‘𝑥))) × ((2nd ‘(1st ‘𝑥))(Hom ‘𝐶)(2nd ‘(2nd ‘𝑥)))) → (1st ‘𝑓) ∈ ((1st ‘(1st ‘𝑥))(Hom ‘𝐴)(1st ‘(2nd ‘𝑥))))
71	69, 70	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (1st ‘𝑓) ∈ ((1st ‘(1st ‘𝑥))(Hom ‘𝐴)(1st ‘(2nd ‘𝑥))))
72		simplr 768	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦))
73	1, 11, 4, 5, 12, 55, 58	xpchom 18197	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦) = (((1st ‘(2nd ‘𝑥))(Hom ‘𝐴)(1st ‘𝑦)) × ((2nd ‘(2nd ‘𝑥))(Hom ‘𝐶)(2nd ‘𝑦))))
74	72, 73	eleqtrd 2837	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → 𝑔 ∈ (((1st ‘(2nd ‘𝑥))(Hom ‘𝐴)(1st ‘𝑦)) × ((2nd ‘(2nd ‘𝑥))(Hom ‘𝐶)(2nd ‘𝑦))))
75		xp1st 8025	. . . . . . . . . 10 ⊢ (𝑔 ∈ (((1st ‘(2nd ‘𝑥))(Hom ‘𝐴)(1st ‘𝑦)) × ((2nd ‘(2nd ‘𝑥))(Hom ‘𝐶)(2nd ‘𝑦))) → (1st ‘𝑔) ∈ ((1st ‘(2nd ‘𝑥))(Hom ‘𝐴)(1st ‘𝑦)))
76	74, 75	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (1st ‘𝑔) ∈ ((1st ‘(2nd ‘𝑥))(Hom ‘𝐴)(1st ‘𝑦)))
77	2, 4, 6, 22, 46, 48, 53, 57, 60, 71, 76	comfeqval 17725	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐴)(1st ‘𝑦))(1st ‘𝑓)) = ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐵)(1st ‘𝑦))(1st ‘𝑓)))
78	28	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (Homf ‘𝐶) = (Homf ‘𝐷))
79		xpcpropd.4	. . . . . . . . . 10 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
80	79	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (compf‘𝐶) = (compf‘𝐷))
81		xp2nd 8026	. . . . . . . . . 10 ⊢ ((1st ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd ‘(1st ‘𝑥)) ∈ (Base‘𝐶))
82	51, 81	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (2nd ‘(1st ‘𝑥)) ∈ (Base‘𝐶))
83		xp2nd 8026	. . . . . . . . . 10 ⊢ ((2nd ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd ‘(2nd ‘𝑥)) ∈ (Base‘𝐶))
84	55, 83	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (2nd ‘(2nd ‘𝑥)) ∈ (Base‘𝐶))
85		xp2nd 8026	. . . . . . . . . 10 ⊢ (𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd ‘𝑦) ∈ (Base‘𝐶))
86	58, 85	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (2nd ‘𝑦) ∈ (Base‘𝐶))
87		xp2nd 8026	. . . . . . . . . 10 ⊢ (𝑓 ∈ (((1st ‘(1st ‘𝑥))(Hom ‘𝐴)(1st ‘(2nd ‘𝑥))) × ((2nd ‘(1st ‘𝑥))(Hom ‘𝐶)(2nd ‘(2nd ‘𝑥)))) → (2nd ‘𝑓) ∈ ((2nd ‘(1st ‘𝑥))(Hom ‘𝐶)(2nd ‘(2nd ‘𝑥))))
88	69, 87	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (2nd ‘𝑓) ∈ ((2nd ‘(1st ‘𝑥))(Hom ‘𝐶)(2nd ‘(2nd ‘𝑥))))
89		xp2nd 8026	. . . . . . . . . 10 ⊢ (𝑔 ∈ (((1st ‘(2nd ‘𝑥))(Hom ‘𝐴)(1st ‘𝑦)) × ((2nd ‘(2nd ‘𝑥))(Hom ‘𝐶)(2nd ‘𝑦))) → (2nd ‘𝑔) ∈ ((2nd ‘(2nd ‘𝑥))(Hom ‘𝐶)(2nd ‘𝑦)))
90	74, 89	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (2nd ‘𝑔) ∈ ((2nd ‘(2nd ‘𝑥))(Hom ‘𝐶)(2nd ‘𝑦)))
91	3, 5, 7, 23, 78, 80, 82, 84, 86, 88, 90	comfeqval 17725	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐶)(2nd ‘𝑦))(2nd ‘𝑓)) = ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐷)(2nd ‘𝑦))(2nd ‘𝑓)))
92	77, 91	opeq12d 4862	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐴)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐶)(2nd ‘𝑦))(2nd ‘𝑓))⟩ = ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐵)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐷)(2nd ‘𝑦))(2nd ‘𝑓))⟩)
93	92	3impa 1109	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐴)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐶)(2nd ‘𝑦))(2nd ‘𝑓))⟩ = ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐵)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐷)(2nd ‘𝑦))(2nd ‘𝑓))⟩)
94	93	mpoeq3dva 7489	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐴)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐶)(2nd ‘𝑦))(2nd ‘𝑓))⟩) = (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐵)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐷)(2nd ‘𝑦))(2nd ‘𝑓))⟩))
95	94	3impa 1109	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐴)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐶)(2nd ‘𝑦))(2nd ‘𝑓))⟩) = (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐵)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐷)(2nd ‘𝑦))(2nd ‘𝑓))⟩))
96	95	mpoeq3dva 7489	. . 3 ⊢ (𝜑 → (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐴)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐶)(2nd ‘𝑦))(2nd ‘𝑓))⟩)) = (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐵)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐷)(2nd ‘𝑦))(2nd ‘𝑓))⟩)))
97	17, 18, 19, 20, 21, 22, 23, 24, 25, 30, 45, 96	xpcval 18194	. 2 ⊢ (𝜑 → (𝐵 ×c 𝐷) = {⟨(Base‘ndx), ((Base‘𝐴) × (Base‘𝐶))⟩, ⟨(Hom ‘ndx), (Hom ‘(𝐴 ×c 𝐶))⟩, ⟨(comp‘ndx), (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝐴)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝐶)(2nd ‘𝑦))(2nd ‘𝑓))⟩))⟩})
98	16, 97	eqtr4d 2774	1 ⊢ (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {ctp 4610  ⟨cop 4612   × cxp 5657  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7991  2^nd c2nd 7992  ndxcnx 17217  Basecbs 17233  Hom chom 17287  compcco 17288  Hom_f chomf 17683  comp_fccomf 17684   ×_c cxpc 18185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-struct 17171  df-slot 17206  df-ndx 17218  df-base 17234  df-hom 17300  df-cco 17301  df-homf 17687  df-comf 17688  df-xpc 18189

This theorem is referenced by:  curfpropd  18250  oppchofcl  18277