Theorem xpccatid 18394

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 18384	. . . 4 ⊢ (𝑋 × 𝑌) = (Base‘𝑇)
5	4	a1i 11	. . 3 ⊢ (𝜑 → (𝑋 × 𝑌) = (Base‘𝑇))
6		eqidd 2764	. . 3 ⊢ (𝜑 → (Hom ‘𝑇) = (Hom ‘𝑇))
7		eqidd 2764	. . 3 ⊢ (𝜑 → (comp‘𝑇) = (comp‘𝑇))
8	1	ovexi 7480	. . . 4 ⊢ 𝑇 ∈ V
9	8	a1i 11	. . 3 ⊢ (𝜑 → 𝑇 ∈ V)
10		biid 264	. . 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 2763	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		xpccat.i	. . . . . 6 ⊢ 𝐼 = (Id‘𝐶)
13		xpccat.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	13	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝐶 ∈ Cat)
15		xp1st 8060	. . . . . . 7 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (1st ‘𝑡) ∈ 𝑋)
16	15	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (1st ‘𝑡) ∈ 𝑋)
17	2, 11, 12, 14, 16	catidcl 17888	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (𝐼‘(1st ‘𝑡)) ∈ ((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑡)))
18		eqid 2763	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
19		xpccat.j	. . . . . 6 ⊢ 𝐽 = (Id‘𝐷)
20		xpccat.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
21	20	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝐷 ∈ Cat)
22		xp2nd 8061	. . . . . . 7 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (2nd ‘𝑡) ∈ 𝑌)
23	22	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (2nd ‘𝑡) ∈ 𝑌)
24	3, 18, 19, 21, 23	catidcl 17888	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (𝐽‘(2nd ‘𝑡)) ∈ ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑡)))
25	17, 24	opelxpd 5695	. . . 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 2763	. . . . 5 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
27		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝑡 ∈ (𝑋 × 𝑌))
28	1, 4, 11, 18, 26, 27, 27	xpchom 18386	. . . 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 2867	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → ⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩ ∈ (𝑡(Hom ‘𝑇)𝑡))
30		fvex 6924	. . . . . . . 8 ⊢ (𝐼‘(1st ‘𝑡)) ∈ V
31		fvex 6924	. . . . . . . 8 ⊢ (𝐽‘(2nd ‘𝑡)) ∈ V
32	30, 31	op1st 8036	. . . . . . 7 ⊢ (1st ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩) = (𝐼‘(1st ‘𝑡))
33	32	oveq1i 7456	. . . . . 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 486	. . . . . . 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 1249	. . . . . . . 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 8060	. . . . . . . 8 ⊢ (𝑠 ∈ (𝑋 × 𝑌) → (1st ‘𝑠) ∈ 𝑋)
37	35, 36	syl 18	. . . . . . 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 2763	. . . . . . 7 ⊢ (comp‘𝐶) = (comp‘𝐶)
39		simpr1r 1250	. . . . . . . 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 18	. . . . . . 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 1282	. . . . . . . . 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 18386	. . . . . . . . 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 2866	. . . . . . . 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 8060	. . . . . . . 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 18	. . . . . . 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 17889	. . . . . 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 2810	. . . . 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 8037	. . . . . . 7 ⊢ (2nd ‘⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩) = (𝐽‘(2nd ‘𝑡))
49	48	oveq1i 7456	. . . . . 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 486	. . . . . . 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 8061	. . . . . . . 8 ⊢ (𝑠 ∈ (𝑋 × 𝑌) → (2nd ‘𝑠) ∈ 𝑌)
52	35, 51	syl 18	. . . . . . 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 2763	. . . . . . 7 ⊢ (comp‘𝐷) = (comp‘𝐷)
54	39, 22	syl 18	. . . . . . 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 8061	. . . . . . . 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 18	. . . . . . 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 17889	. . . . . 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 2810	. . . . 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 4851	. . . 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 2763	. . . . 5 ⊢ (comp‘𝑇) = (comp‘𝑇)
61	39, 29	syldan 603	. . . . 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 18389	. . . 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 8067	. . . . 5 ⊢ (𝑓 ∈ (((1st ‘𝑠)(Hom ‘𝐶)(1st ‘𝑡)) × ((2nd ‘𝑠)(Hom ‘𝐷)(2nd ‘𝑡))) → 𝑓 = ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)
64	43, 63	syl 18	. . . 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 2808	. . 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 7457	. . . . . 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 1251	. . . . . . . 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 8060	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (1st ‘𝑢) ∈ 𝑋)
69	67, 68	syl 18	. . . . . . 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 1283	. . . . . . . . 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 18386	. . . . . . . . 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 2866	. . . . . . . 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 8060	. . . . . . . 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 18	. . . . . . 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 17890	. . . . . 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 2810	. . . . 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 7457	. . . . . 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 8061	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (2nd ‘𝑢) ∈ 𝑌)
79	67, 78	syl 18	. . . . . . 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 8061	. . . . . . . 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 18	. . . . . . 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 17890	. . . . . 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 2810	. . . . 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 4851	. . . 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 18389	. . . 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 8067	. . . . 5 ⊢ (𝑔 ∈ (((1st ‘𝑡)(Hom ‘𝐶)(1st ‘𝑢)) × ((2nd ‘𝑡)(Hom ‘𝐷)(2nd ‘𝑢))) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
87	72, 86	syl 18	. . . 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 2808	. . 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 17891	. . . . 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 17891	. . . . 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 5695	. . . 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 18389	. . . 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 18386	. . . 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 2879	. . 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 1252	. . . . . . . 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 8060	. . . . . . . 8 ⊢ (𝑣 ∈ (𝑋 × 𝑌) → (1st ‘𝑣) ∈ 𝑋)
97	95, 96	syl 18	. . . . . . 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 1284	. . . . . . . . 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 18386	. . . . . . . . 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 2866	. . . . . . . 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 8060	. . . . . . . 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 18	. . . . . . 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 17892	. . . . . 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 18389	. . . . . . . . 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 6915	. . . . . . . 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 7479	. . . . . . . . 9 ⊢ ((1st ‘ℎ)(⟨(1st ‘𝑡), (1st ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑣))(1st ‘𝑔)) ∈ V
107		ovex 7479	. . . . . . . . 9 ⊢ ((2nd ‘ℎ)(⟨(2nd ‘𝑡), (2nd ‘𝑢)⟩(comp‘𝐷)(2nd ‘𝑣))(2nd ‘𝑔)) ∈ V
108	106, 107	op1st 8036	. . . . . . . 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 2814	. . . . . . 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 7461	. . . . . 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 6915	. . . . . . . 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 7479	. . . . . . . . 9 ⊢ ((1st ‘𝑔)(⟨(1st ‘𝑠), (1st ‘𝑡)⟩(comp‘𝐶)(1st ‘𝑢))(1st ‘𝑓)) ∈ V
113		ovex 7479	. . . . . . . . 9 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘𝑠), (2nd ‘𝑡)⟩(comp‘𝐷)(2nd ‘𝑢))(2nd ‘𝑓)) ∈ V
114	112, 113	op1st 8036	. . . . . . . 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 2814	. . . . . . 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 7462	. . . . . 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 2808	. . . . 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 8061	. . . . . . . 8 ⊢ (𝑣 ∈ (𝑋 × 𝑌) → (2nd ‘𝑣) ∈ 𝑌)
119	95, 118	syl 18	. . . . . . 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 8061	. . . . . . . 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 18	. . . . . . 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 17892	. . . . . 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 6915	. . . . . . . 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 8037	. . . . . . . 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 2814	. . . . . . 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 7461	. . . . . 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 6915	. . . . . . . 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 8037	. . . . . . . 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 2814	. . . . . . 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 7462	. . . . . 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 2808	. . . . 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 4851	. . . 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 17891	. . . . . . 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 17891	. . . . . . 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 5695	. . . . . 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 18386	. . . . . 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 2879	. . . . 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 18389	. . . 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 18389	. . . 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 2808	. . 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 17887	. 2 ⊢ (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑡 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩)))
142		vex 3458	. . . . . . . 8 ⊢ 𝑥 ∈ V
143		vex 3458	. . . . . . . 8 ⊢ 𝑦 ∈ V
144	142, 143	op1std 8038	. . . . . . 7 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑡) = 𝑥)
145	144	fveq2d 6915	. . . . . 6 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝐼‘(1st ‘𝑡)) = (𝐼‘𝑥))
146	142, 143	op2ndd 8039	. . . . . . 7 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑡) = 𝑦)
147	146	fveq2d 6915	. . . . . 6 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝐽‘(2nd ‘𝑡)) = (𝐽‘𝑦))
148	145, 147	opeq12d 4851	. . . . 5 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → ⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩ = ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)
149	148	mpompt 7560	. . . 4 ⊢ (𝑡 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐼‘(1st ‘𝑡)), (𝐽‘(2nd ‘𝑡))⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)
150	149	eqeq2i 2776	. . 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 635	. 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 221	1 ⊢ (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4838   ↦ cmpt 5190   × cxp 5654  ‘cfv 6561  (class class class)co 7446   ∈ cmpo 7448  1^st c1st 8026  2^nd c2nd 8027  Basecbs 17419  Hom chom 17471  compcco 17472  Catccat 17870  Idccid 17871   ×_c cxpc 18374

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-er 8748  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-2 12435  df-3 12436  df-4 12437  df-5 12438  df-6 12439  df-7 12440  df-8 12441  df-9 12442  df-n0 12637  df-z 12724  df-dec 12845  df-uz 12996  df-fz 13670  df-struct 17357  df-slot 17392  df-ndx 17404  df-base 17420  df-hom 17484  df-cco 17485  df-cat 17874  df-cid 17875  df-xpc 18378

This theorem is used by:  xpcid  18395  xpccat  18396