Theorem xpccatid 18152

Description: The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
xpccat.t	⊢ 𝑇 = (𝐶 ×c 𝐷)
xpccat.c	⊢ (𝜑 → 𝐶 ∈ Cat)
xpccat.d	⊢ (𝜑 → 𝐷 ∈ Cat)
xpccat.x	⊢ 𝑋 = (Base‘𝐶)
xpccat.y	⊢ 𝑌 = (Base‘𝐷)
xpccat.i	⊢ 𝐼 = (Id‘𝐶)
xpccat.j	⊢ 𝐽 = (Id‘𝐷)

Assertion

Ref	Expression
xpccatid	⊢ (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)))

Distinct variable groups:   𝑥,𝑦,𝐼   𝑥,𝐽,𝑦   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝐷,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝑇(𝑥,𝑦)

Proof of Theorem xpccatid

Dummy variables 𝑓 𝑔 ℎ 𝑠 𝑡 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpccat.t	. . . . 5 ⊢ 𝑇 = (𝐶 ×c 𝐷)
2		xpccat.x	. . . . 5 ⊢ 𝑋 = (Base‘𝐶)
3		xpccat.y	. . . . 5 ⊢ 𝑌 = (Base‘𝐷)
4	1, 2, 3	xpcbas 18142	. . . 4 ⊢ (𝑋 × 𝑌) = (Base‘𝑇)
5	4	a1i 11	. . 3 ⊢ (𝜑 → (𝑋 × 𝑌) = (Base‘𝑇))
6		eqidd 2727	. . 3 ⊢ (𝜑 → (Hom ‘𝑇) = (Hom ‘𝑇))
7		eqidd 2727	. . 3 ⊢ (𝜑 → (comp‘𝑇) = (comp‘𝑇))
8	1	ovexi 7439	. . . 4 ⊢ 𝑇 ∈ V
9	8	a1i 11	. . 3 ⊢ (𝜑 → 𝑇 ∈ V)
10		biid 261	. . 3 ⊢ (((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣))) ↔ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣))))
11		eqid 2726	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		xpccat.i	. . . . . 6 ⊢ 𝐼 = (Id‘𝐶)
13		xpccat.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝐶 ∈ Cat)
15		xp1st 8006	. . . . . . 7 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (1st ‘𝑡) ∈ 𝑋)
16	15	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (1st ‘𝑡) ∈ 𝑋)
17	2, 11, 12, 14, 16	catidcl 17635	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (𝐼‘(1st ‘𝑡)) ∈ ((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑡)))
18		eqid 2726	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
19		xpccat.j	. . . . . 6 ⊢ 𝐽 = (Id‘𝐷)
20		xpccat.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
21	20	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝐷 ∈ Cat)
22		xp2nd 8007	. . . . . . 7 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (2nd ‘𝑡) ∈ 𝑌)
23	22	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (2nd ‘𝑡) ∈ 𝑌)
24	3, 18, 19, 21, 23	catidcl 17635	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (𝐽‘(2nd ‘𝑡)) ∈ ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑡)))
25	17, 24	opelxpd 5708	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → ⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩ ∈ (((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑡)) × ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑡))))
26		eqid 2726	. . . . 5 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
27		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝑡 ∈ (𝑋 × 𝑌))
28	1, 4, 11, 18, 26, 27, 27	xpchom 18144	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (𝑡(Hom ‘𝑇)𝑡) = (((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑡)) × ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑡))))
29	25, 28	eleqtrrd 2830	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → ⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩ ∈ (𝑡(Hom ‘𝑇)𝑡))
30		fvex 6898	. . . . . . . 8 ⊢ (𝐼‘(1st ‘𝑡)) ∈ V
31		fvex 6898	. . . . . . . 8 ⊢ (𝐽‘(2nd ‘𝑡)) ∈ V
32	30, 31	op1st 7982	. . . . . . 7 ⊢ (1st ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩) = (𝐼‘(1st ‘𝑡))
33	32	oveq1i 7415	. . . . . 6 ⊢ ((1st ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑡))(1st ‘𝑓)) = ((𝐼‘(1st ‘𝑡))(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑡))(1st ‘𝑓))
34	13	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝐶 ∈ Cat)
35		simpr1l 1227	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑠 ∈ (𝑋 × 𝑌))
36		xp1st 8006	. . . . . . . 8 ⊢ (𝑠 ∈ (𝑋 × 𝑌) → (1st ‘𝑠) ∈ 𝑋)
37	35, 36	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘𝑠) ∈ 𝑋)
38		eqid 2726	. . . . . . 7 ⊢ (comp‘𝐶) = (comp‘𝐶)
39		simpr1r 1228	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑡 ∈ (𝑋 × 𝑌))
40	39, 15	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘𝑡) ∈ 𝑋)
41		simpr31 1260	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡))
42	1, 4, 11, 18, 26, 35, 39	xpchom 18144	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑠(Hom ‘𝑇)𝑡) = (((1st ‘𝑠)(Hom ‘𝐶)(1st ‘𝑡)) × ((2nd ‘𝑠)(Hom ‘𝐷)(2nd ‘𝑡))))
43	41, 42	eleqtrd 2829	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑓 ∈ (((1st ‘𝑠)(Hom ‘𝐶)(1st ‘𝑡)) × ((2nd ‘𝑠)(Hom ‘𝐷)(2nd ‘𝑡))))
44		xp1st 8006	. . . . . . . 8 ⊢ (𝑓 ∈ (((1st ‘𝑠)(Hom ‘𝐶)(1st ‘𝑡)) × ((2nd ‘𝑠)(Hom ‘𝐷)(2nd ‘𝑡))) → (1st ‘𝑓) ∈ ((1st ‘𝑠)(Hom ‘𝐶)(1st ‘𝑡)))
45	43, 44	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘𝑓) ∈ ((1st ‘𝑠)(Hom ‘𝐶)(1st ‘𝑡)))
46	2, 11, 12, 34, 37, 38, 40, 45	catlid 17636	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((𝐼‘(1st ‘𝑡))(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑡))(1st ‘𝑓)) = (1st ‘𝑓))
47	33, 46	eqtrid 2778	. . . . 5 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑡))(1st ‘𝑓)) = (1st ‘𝑓))
48	30, 31	op2nd 7983	. . . . . . 7 ⊢ (2nd ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩) = (𝐽‘(2nd ‘𝑡))
49	48	oveq1i 7415	. . . . . 6 ⊢ ((2nd ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑡))(2nd ‘𝑓)) = ((𝐽‘(2nd ‘𝑡))(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑡))(2nd ‘𝑓))
50	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝐷 ∈ Cat)
51		xp2nd 8007	. . . . . . . 8 ⊢ (𝑠 ∈ (𝑋 × 𝑌) → (2nd ‘𝑠) ∈ 𝑌)
52	35, 51	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘𝑠) ∈ 𝑌)
53		eqid 2726	. . . . . . 7 ⊢ (comp‘𝐷) = (comp‘𝐷)
54	39, 22	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘𝑡) ∈ 𝑌)
55		xp2nd 8007	. . . . . . . 8 ⊢ (𝑓 ∈ (((1st ‘𝑠)(Hom ‘𝐶)(1st ‘𝑡)) × ((2nd ‘𝑠)(Hom ‘𝐷)(2nd ‘𝑡))) → (2nd ‘𝑓) ∈ ((2nd ‘𝑠)(Hom ‘𝐷)(2nd ‘𝑡)))
56	43, 55	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘𝑓) ∈ ((2nd ‘𝑠)(Hom ‘𝐷)(2nd ‘𝑡)))
57	3, 18, 19, 50, 52, 53, 54, 56	catlid 17636	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((𝐽‘(2nd ‘𝑡))(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑡))(2nd ‘𝑓)) = (2nd ‘𝑓))
58	49, 57	eqtrid 2778	. . . . 5 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑡))(2nd ‘𝑓)) = (2nd ‘𝑓))
59	47, 58	opeq12d 4876	. . . 4 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ⟨((1st ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑡))(1st ‘𝑓)), ((2nd ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑡))(2nd ‘𝑓))⟩ = ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)
60		eqid 2726	. . . . 5 ⊢ (comp‘𝑇) = (comp‘𝑇)
61	39, 29	syldan 590	. . . . 5 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩ ∈ (𝑡(Hom ‘𝑇)𝑡))
62	1, 4, 26, 38, 53, 60, 35, 39, 39, 41, 61	xpcco 18147	. . . 4 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑡)𝑓) = ⟨((1st ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑡))(1st ‘𝑓)), ((2nd ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑡))(2nd ‘𝑓))⟩)
63		1st2nd2 8013	. . . . 5 ⊢ (𝑓 ∈ (((1st ‘𝑠)(Hom ‘𝐶)(1st ‘𝑡)) × ((2nd ‘𝑠)(Hom ‘𝐷)(2nd ‘𝑡))) → 𝑓 = ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)
64	43, 63	syl 17	. . . 4 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑓 = ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)
65	59, 62, 64	3eqtr4d 2776	. . 3 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑡)𝑓) = 𝑓)
66	32	oveq2i 7416	. . . . . 6 ⊢ ((1st ‘𝑔)(⟨(1st ‘𝑡), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)) = ((1st ‘𝑔)(⟨(1st ‘𝑡), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(𝐼‘(1st ‘𝑡)))
67		simpr2l 1229	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑢 ∈ (𝑋 × 𝑌))
68		xp1st 8006	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (1st ‘𝑢) ∈ 𝑋)
69	67, 68	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘𝑢) ∈ 𝑋)
70		simpr32 1261	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢))
71	1, 4, 11, 18, 26, 39, 67	xpchom 18144	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑡(Hom ‘𝑇)𝑢) = (((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑢)) × ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑢))))
72	70, 71	eleqtrd 2829	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑔 ∈ (((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑢)) × ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑢))))
73		xp1st 8006	. . . . . . . 8 ⊢ (𝑔 ∈ (((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑢)) × ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑢))) → (1st ‘𝑔) ∈ ((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑢)))
74	72, 73	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘𝑔) ∈ ((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑢)))
75	2, 11, 12, 34, 40, 38, 69, 74	catrid 17637	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st ‘𝑔)(⟨(1st ‘𝑡), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(𝐼‘(1st ‘𝑡))) = (1st ‘𝑔))
76	66, 75	eqtrid 2778	. . . . 5 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st ‘𝑔)(⟨(1st ‘𝑡), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)) = (1st ‘𝑔))
77	48	oveq2i 7416	. . . . . 6 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘𝑡), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)) = ((2nd ‘𝑔)(⟨(2nd ‘𝑡), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(𝐽‘(2nd ‘𝑡)))
78		xp2nd 8007	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (2nd ‘𝑢) ∈ 𝑌)
79	67, 78	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘𝑢) ∈ 𝑌)
80		xp2nd 8007	. . . . . . . 8 ⊢ (𝑔 ∈ (((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑢)) × ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑢))) → (2nd ‘𝑔) ∈ ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑢)))
81	72, 80	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘𝑔) ∈ ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑢)))
82	3, 18, 19, 50, 54, 53, 79, 81	catrid 17637	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd ‘𝑔)(⟨(2nd ‘𝑡), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(𝐽‘(2nd ‘𝑡))) = (2nd ‘𝑔))
83	77, 82	eqtrid 2778	. . . . 5 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd ‘𝑔)(⟨(2nd ‘𝑡), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)) = (2nd ‘𝑔))
84	76, 83	opeq12d 4876	. . . 4 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ⟨((1st ‘𝑔)(⟨(1st ‘𝑡), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)), ((2nd ‘𝑔)(⟨(2nd ‘𝑡), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩))⟩ = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
85	1, 4, 26, 38, 53, 60, 39, 39, 67, 61, 70	xpcco 18147	. . . 4 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑔(⟨𝑡, 𝑡⟩(comp‘𝑇)𝑢)⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩) = ⟨((1st ‘𝑔)(⟨(1st ‘𝑡), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)), ((2nd ‘𝑔)(⟨(2nd ‘𝑡), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩))⟩)
86		1st2nd2 8013	. . . . 5 ⊢ (𝑔 ∈ (((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑢)) × ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑢))) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
87	72, 86	syl 17	. . . 4 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
88	84, 85, 87	3eqtr4d 2776	. . 3 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑔(⟨𝑡, 𝑡⟩(comp‘𝑇)𝑢)⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩) = 𝑔)
89	2, 11, 38, 34, 37, 40, 69, 45, 74	catcocl 17638	. . . . 5 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st ‘𝑔)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘𝑓)) ∈ ((1st ‘𝑠)(Hom ‘𝐶)(1st ‘𝑢)))
90	3, 18, 53, 50, 52, 54, 79, 56, 81	catcocl 17638	. . . . 5 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd ‘𝑔)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘𝑓)) ∈ ((2nd ‘𝑠)(Hom ‘𝐷)(2nd ‘𝑢)))
91	89, 90	opelxpd 5708	. . . 4 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ⟨((1st ‘𝑔)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘𝑓))⟩ ∈ (((1st ‘𝑠)(Hom ‘𝐶)(1st ‘𝑢)) × ((2nd ‘𝑠)(Hom ‘𝐷)(2nd ‘𝑢))))
92	1, 4, 26, 38, 53, 60, 35, 39, 67, 41, 70	xpcco 18147	. . . 4 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓) = ⟨((1st ‘𝑔)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘𝑓))⟩)
93	1, 4, 11, 18, 26, 35, 67	xpchom 18144	. . . 4 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑠(Hom ‘𝑇)𝑢) = (((1st ‘𝑠)(Hom ‘𝐶)(1st ‘𝑢)) × ((2nd ‘𝑠)(Hom ‘𝐷)(2nd ‘𝑢))))
94	91, 92, 93	3eltr4d 2842	. . 3 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓) ∈ (𝑠(Hom ‘𝑇)𝑢))
95		simpr2r 1230	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑣 ∈ (𝑋 × 𝑌))
96		xp1st 8006	. . . . . . . 8 ⊢ (𝑣 ∈ (𝑋 × 𝑌) → (1st ‘𝑣) ∈ 𝑋)
97	95, 96	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘𝑣) ∈ 𝑋)
98		simpr33 1262	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ℎ ∈ (𝑢(Hom ‘𝑇)𝑣))
99	1, 4, 11, 18, 26, 67, 95	xpchom 18144	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑢(Hom ‘𝑇)𝑣) = (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣))))
100	98, 99	eleqtrd 2829	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ℎ ∈ (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣))))
101		xp1st 8006	. . . . . . . 8 ⊢ (ℎ ∈ (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣))) → (1st ‘ℎ) ∈ ((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)))
102	100, 101	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘ℎ) ∈ ((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)))
103	2, 11, 38, 34, 37, 40, 69, 45, 74, 97, 102	catass 17639	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (((1st ‘ℎ)(⟨(1st ‘𝑡), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑔))(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑓)) = ((1st ‘ℎ)(⟨(1st ‘𝑠), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))((1st ‘𝑔)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘𝑓))))
104	1, 4, 26, 38, 53, 60, 39, 67, 95, 70, 98	xpcco 18147	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔) = ⟨((1st ‘ℎ)(⟨(1st ‘𝑡), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑔)), ((2nd ‘ℎ)(⟨(2nd ‘𝑡), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑔))⟩)
105	104	fveq2d 6889	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)) = (1st ‘⟨((1st ‘ℎ)(⟨(1st ‘𝑡), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑔)), ((2nd ‘ℎ)(⟨(2nd ‘𝑡), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑔))⟩))
106		ovex 7438	. . . . . . . . 9 ⊢ ((1st ‘ℎ)(⟨(1st ‘𝑡), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑔)) ∈ V
107		ovex 7438	. . . . . . . . 9 ⊢ ((2nd ‘ℎ)(⟨(2nd ‘𝑡), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑔)) ∈ V
108	106, 107	op1st 7982	. . . . . . . 8 ⊢ (1st ‘⟨((1st ‘ℎ)(⟨(1st ‘𝑡), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑔)), ((2nd ‘ℎ)(⟨(2nd ‘𝑡), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑔))⟩) = ((1st ‘ℎ)(⟨(1st ‘𝑡), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑔))
109	105, 108	eqtrdi 2782	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)) = ((1st ‘ℎ)(⟨(1st ‘𝑡), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑔)))
110	109	oveq1d 7420	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑓)) = (((1st ‘ℎ)(⟨(1st ‘𝑡), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑔))(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑓)))
111	92	fveq2d 6889	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)) = (1st ‘⟨((1st ‘𝑔)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘𝑓))⟩))
112		ovex 7438	. . . . . . . . 9 ⊢ ((1st ‘𝑔)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘𝑓)) ∈ V
113		ovex 7438	. . . . . . . . 9 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘𝑓)) ∈ V
114	112, 113	op1st 7982	. . . . . . . 8 ⊢ (1st ‘⟨((1st ‘𝑔)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘𝑓))⟩) = ((1st ‘𝑔)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘𝑓))
115	111, 114	eqtrdi 2782	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)) = ((1st ‘𝑔)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘𝑓)))
116	115	oveq2d 7421	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st ‘ℎ)(⟨(1st ‘𝑠), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))) = ((1st ‘ℎ)(⟨(1st ‘𝑠), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))((1st ‘𝑔)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘𝑓))))
117	103, 110, 116	3eqtr4d 2776	. . . . 5 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑓)) = ((1st ‘ℎ)(⟨(1st ‘𝑠), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))))
118		xp2nd 8007	. . . . . . . 8 ⊢ (𝑣 ∈ (𝑋 × 𝑌) → (2nd ‘𝑣) ∈ 𝑌)
119	95, 118	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘𝑣) ∈ 𝑌)
120		xp2nd 8007	. . . . . . . 8 ⊢ (ℎ ∈ (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣))) → (2nd ‘ℎ) ∈ ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣)))
121	100, 120	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘ℎ) ∈ ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣)))
122	3, 18, 53, 50, 52, 54, 79, 56, 81, 119, 121	catass 17639	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (((2nd ‘ℎ)(⟨(2nd ‘𝑡), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑔))(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑓)) = ((2nd ‘ℎ)(⟨(2nd ‘𝑠), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))((2nd ‘𝑔)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘𝑓))))
123	104	fveq2d 6889	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)) = (2nd ‘⟨((1st ‘ℎ)(⟨(1st ‘𝑡), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑔)), ((2nd ‘ℎ)(⟨(2nd ‘𝑡), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑔))⟩))
124	106, 107	op2nd 7983	. . . . . . . 8 ⊢ (2nd ‘⟨((1st ‘ℎ)(⟨(1st ‘𝑡), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑔)), ((2nd ‘ℎ)(⟨(2nd ‘𝑡), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑔))⟩) = ((2nd ‘ℎ)(⟨(2nd ‘𝑡), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑔))
125	123, 124	eqtrdi 2782	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)) = ((2nd ‘ℎ)(⟨(2nd ‘𝑡), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑔)))
126	125	oveq1d 7420	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑓)) = (((2nd ‘ℎ)(⟨(2nd ‘𝑡), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑔))(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑓)))
127	92	fveq2d 6889	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)) = (2nd ‘⟨((1st ‘𝑔)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘𝑓))⟩))
128	112, 113	op2nd 7983	. . . . . . . 8 ⊢ (2nd ‘⟨((1st ‘𝑔)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘𝑓))⟩) = ((2nd ‘𝑔)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘𝑓))
129	127, 128	eqtrdi 2782	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)) = ((2nd ‘𝑔)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘𝑓)))
130	129	oveq2d 7421	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd ‘ℎ)(⟨(2nd ‘𝑠), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))) = ((2nd ‘ℎ)(⟨(2nd ‘𝑠), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))((2nd ‘𝑔)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘𝑓))))
131	122, 126, 130	3eqtr4d 2776	. . . . 5 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑓)) = ((2nd ‘ℎ)(⟨(2nd ‘𝑠), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))))
132	117, 131	opeq12d 4876	. . . 4 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ⟨((1st ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑓)), ((2nd ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑓))⟩ = ⟨((1st ‘ℎ)(⟨(1st ‘𝑠), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))), ((2nd ‘ℎ)(⟨(2nd ‘𝑠), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)))⟩)
133	2, 11, 38, 34, 40, 69, 97, 74, 102	catcocl 17638	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st ‘ℎ)(⟨(1st ‘𝑡), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑔)) ∈ ((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑣)))
134	3, 18, 53, 50, 54, 79, 119, 81, 121	catcocl 17638	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd ‘ℎ)(⟨(2nd ‘𝑡), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑔)) ∈ ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑣)))
135	133, 134	opelxpd 5708	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ⟨((1st ‘ℎ)(⟨(1st ‘𝑡), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑔)), ((2nd ‘ℎ)(⟨(2nd ‘𝑡), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑔))⟩ ∈ (((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑣))))
136	1, 4, 11, 18, 26, 39, 95	xpchom 18144	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑡(Hom ‘𝑇)𝑣) = (((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑣))))
137	135, 104, 136	3eltr4d 2842	. . . . 5 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔) ∈ (𝑡(Hom ‘𝑇)𝑣))
138	1, 4, 26, 38, 53, 60, 35, 39, 95, 41, 137	xpcco 18147	. . . 4 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑣)𝑓) = ⟨((1st ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑓)), ((2nd ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑓))⟩)
139	1, 4, 26, 38, 53, 60, 35, 67, 95, 94, 98	xpcco 18147	. . . 4 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (ℎ(⟨𝑠, 𝑢⟩(comp‘𝑇)𝑣)(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)) = ⟨((1st ‘ℎ)(⟨(1st ‘𝑠), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))), ((2nd ‘ℎ)(⟨(2nd ‘𝑠), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)))⟩)
140	132, 138, 139	3eqtr4d 2776	. . 3 ⊢ ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ℎ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑣)𝑓) = (ℎ(⟨𝑠, 𝑢⟩(comp‘𝑇)𝑣)(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)))
141	5, 6, 7, 9, 10, 29, 65, 88, 94, 140	iscatd2 17634	. 2 ⊢ (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑡 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)))
142		vex 3472	. . . . . . . 8 ⊢ 𝑥 ∈ V
143		vex 3472	. . . . . . . 8 ⊢ 𝑦 ∈ V
144	142, 143	op1std 7984	. . . . . . 7 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑡) = 𝑥)
145	144	fveq2d 6889	. . . . . 6 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝐼‘(1st ‘𝑡)) = (𝐼‘𝑥))
146	142, 143	op2ndd 7985	. . . . . . 7 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑡) = 𝑦)
147	146	fveq2d 6889	. . . . . 6 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝐽‘(2nd ‘𝑡)) = (𝐽‘𝑦))
148	145, 147	opeq12d 4876	. . . . 5 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → ⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩ = ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)
149	148	mpompt 7518	. . . 4 ⊢ (𝑡 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)
150	149	eqeq2i 2739	. . 3 ⊢ ((Id‘𝑇) = (𝑡 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩) ↔ (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩))
151	150	anbi2i 622	. 2 ⊢ ((𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑡 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)) ↔ (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)))
152	141, 151	sylib 217	1 ⊢ (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3468  ⟨cop 4629   ↦ cmpt 5224   × cxp 5667  ‘cfv 6537  (class class class)co 7405   ∈ cmpo 7407  1^st c1st 7972  2^nd c2nd 7973  Basecbs 17153  Hom chom 17217  compcco 17218  Catccat 17617  Idccid 17618   ×_c cxpc 18132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13491  df-struct 17089  df-slot 17124  df-ndx 17136  df-base 17154  df-hom 17230  df-cco 17231  df-cat 17621  df-cid 17622  df-xpc 18136

This theorem is referenced by:  xpcid  18153  xpccat  18154