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Theorem xpccatid 18096

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 18086	. . . 4 ⊢ (𝑋 × 𝑌) = (Base‘𝑇)
5	4	a1i 11	. . 3 ⊢ (𝜑 → (𝑋 × 𝑌) = (Base‘𝑇))
6		eqidd 2734	. . 3 ⊢ (𝜑 → (Hom ‘𝑇) = (Hom ‘𝑇))
7		eqidd 2734	. . 3 ⊢ (𝜑 → (comp‘𝑇) = (comp‘𝑇))
8	1	ovexi 7386	. . . 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 2733	. . . . . 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 7959	. . . . . . 7 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (1^st ‘𝑡) ∈ 𝑋)
16	15	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (1^st ‘𝑡) ∈ 𝑋)
17	2, 11, 12, 14, 16	catidcl 17590	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (𝐼‘(1^st ‘𝑡)) ∈ ((1^st ‘𝑡)(Hom ‘𝐶)(1^st ‘𝑡)))
18		eqid 2733	. . . . . 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 7960	. . . . . . 7 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑡) ∈ 𝑌)
23	22	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (2^nd ‘𝑡) ∈ 𝑌)
24	3, 18, 19, 21, 23	catidcl 17590	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (𝐽‘(2^nd ‘𝑡)) ∈ ((2^nd ‘𝑡)(Hom ‘𝐷)(2^nd ‘𝑡)))
25	17, 24	opelxpd 5658	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → ⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩ ∈ (((1^st ‘𝑡)(Hom ‘𝐶)(1^st ‘𝑡)) × ((2^nd ‘𝑡)(Hom ‘𝐷)(2^nd ‘𝑡))))
26		eqid 2733	. . . . 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 18088	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (𝑡(Hom ‘𝑇)𝑡) = (((1^st ‘𝑡)(Hom ‘𝐶)(1^st ‘𝑡)) × ((2^nd ‘𝑡)(Hom ‘𝐷)(2^nd ‘𝑡))))
29	25, 28	eleqtrrd 2836	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → ⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩ ∈ (𝑡(Hom ‘𝑇)𝑡))
30		fvex 6841	. . . . . . . 8 ⊢ (𝐼‘(1^st ‘𝑡)) ∈ V
31		fvex 6841	. . . . . . . 8 ⊢ (𝐽‘(2^nd ‘𝑡)) ∈ V
32	30, 31	op1st 7935	. . . . . . 7 ⊢ (1^st ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩) = (𝐼‘(1^st ‘𝑡))
33	32	oveq1i 7362	. . . . . 6 ⊢ ((1^st ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑡))(1^st ‘𝑓)) = ((𝐼‘(1^st ‘𝑡))(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑡))(1^st ‘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 1231	. . . . . . . 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 7959	. . . . . . . 8 ⊢ (𝑠 ∈ (𝑋 × 𝑌) → (1^st ‘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)))) → (1^st ‘𝑠) ∈ 𝑋)
38		eqid 2733	. . . . . . 7 ⊢ (comp‘𝐶) = (comp‘𝐶)
39		simpr1r 1232	. . . . . . . 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)))) → (1^st ‘𝑡) ∈ 𝑋)
41		simpr31 1264	. . . . . . . . 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 18088	. . . . . . . . 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) = (((1^st ‘𝑠)(Hom ‘𝐶)(1^st ‘𝑡)) × ((2^nd ‘𝑠)(Hom ‘𝐷)(2^nd ‘𝑡))))
43	41, 42	eleqtrd 2835	. . . . . . . 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 ∈ (((1^st ‘𝑠)(Hom ‘𝐶)(1^st ‘𝑡)) × ((2^nd ‘𝑠)(Hom ‘𝐷)(2^nd ‘𝑡))))
44		xp1st 7959	. . . . . . . 8 ⊢ (𝑓 ∈ (((1^st ‘𝑠)(Hom ‘𝐶)(1^st ‘𝑡)) × ((2^nd ‘𝑠)(Hom ‘𝐷)(2^nd ‘𝑡))) → (1^st ‘𝑓) ∈ ((1^st ‘𝑠)(Hom ‘𝐶)(1^st ‘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)))) → (1^st ‘𝑓) ∈ ((1^st ‘𝑠)(Hom ‘𝐶)(1^st ‘𝑡)))
46	2, 11, 12, 34, 37, 38, 40, 45	catlid 17591	. . . . . 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‘(1^st ‘𝑡))(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑡))(1^st ‘𝑓)) = (1^st ‘𝑓))
47	33, 46	eqtrid 2780	. . . . 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)))) → ((1^st ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑡))(1^st ‘𝑓)) = (1^st ‘𝑓))
48	30, 31	op2nd 7936	. . . . . . 7 ⊢ (2^nd ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩) = (𝐽‘(2^nd ‘𝑡))
49	48	oveq1i 7362	. . . . . 6 ⊢ ((2^nd ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑡))(2^nd ‘𝑓)) = ((𝐽‘(2^nd ‘𝑡))(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑡))(2^nd ‘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 7960	. . . . . . . 8 ⊢ (𝑠 ∈ (𝑋 × 𝑌) → (2^nd ‘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)))) → (2^nd ‘𝑠) ∈ 𝑌)
53		eqid 2733	. . . . . . 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)))) → (2^nd ‘𝑡) ∈ 𝑌)
55		xp2nd 7960	. . . . . . . 8 ⊢ (𝑓 ∈ (((1^st ‘𝑠)(Hom ‘𝐶)(1^st ‘𝑡)) × ((2^nd ‘𝑠)(Hom ‘𝐷)(2^nd ‘𝑡))) → (2^nd ‘𝑓) ∈ ((2^nd ‘𝑠)(Hom ‘𝐷)(2^nd ‘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)))) → (2^nd ‘𝑓) ∈ ((2^nd ‘𝑠)(Hom ‘𝐷)(2^nd ‘𝑡)))
57	3, 18, 19, 50, 52, 53, 54, 56	catlid 17591	. . . . . 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‘(2^nd ‘𝑡))(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑡))(2^nd ‘𝑓)) = (2^nd ‘𝑓))
58	49, 57	eqtrid 2780	. . . . 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)))) → ((2^nd ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑡))(2^nd ‘𝑓)) = (2^nd ‘𝑓))
59	47, 58	opeq12d 4832	. . . 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)))) → ⟨((1^st ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑡))(1^st ‘𝑓)), ((2^nd ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑡))(2^nd ‘𝑓))⟩ = ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩)
60		eqid 2733	. . . . 5 ⊢ (comp‘𝑇) = (comp‘𝑇)
61	39, 29	syldan 591	. . . . 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‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩ ∈ (𝑡(Hom ‘𝑇)𝑡))
62	1, 4, 26, 38, 53, 60, 35, 39, 39, 41, 61	xpcco 18091	. . . 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‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑡)𝑓) = ⟨((1^st ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑡))(1^st ‘𝑓)), ((2^nd ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑡))(2^nd ‘𝑓))⟩)
63		1st2nd2 7966	. . . . 5 ⊢ (𝑓 ∈ (((1^st ‘𝑠)(Hom ‘𝐶)(1^st ‘𝑡)) × ((2^nd ‘𝑠)(Hom ‘𝐷)(2^nd ‘𝑡))) → 𝑓 = ⟨(1^st ‘𝑓), (2^nd ‘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 = ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩)
65	59, 62, 64	3eqtr4d 2778	. . 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‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑡)𝑓) = 𝑓)
66	32	oveq2i 7363	. . . . . 6 ⊢ ((1^st ‘𝑔)(⟨(1^st ‘𝑡), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)) = ((1^st ‘𝑔)(⟨(1^st ‘𝑡), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(𝐼‘(1^st ‘𝑡)))
67		simpr2l 1233	. . . . . . . 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 7959	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (1^st ‘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)))) → (1^st ‘𝑢) ∈ 𝑋)
70		simpr32 1265	. . . . . . . . 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 18088	. . . . . . . . 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) = (((1^st ‘𝑡)(Hom ‘𝐶)(1^st ‘𝑢)) × ((2^nd ‘𝑡)(Hom ‘𝐷)(2^nd ‘𝑢))))
72	70, 71	eleqtrd 2835	. . . . . . . 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 ∈ (((1^st ‘𝑡)(Hom ‘𝐶)(1^st ‘𝑢)) × ((2^nd ‘𝑡)(Hom ‘𝐷)(2^nd ‘𝑢))))
73		xp1st 7959	. . . . . . . 8 ⊢ (𝑔 ∈ (((1^st ‘𝑡)(Hom ‘𝐶)(1^st ‘𝑢)) × ((2^nd ‘𝑡)(Hom ‘𝐷)(2^nd ‘𝑢))) → (1^st ‘𝑔) ∈ ((1^st ‘𝑡)(Hom ‘𝐶)(1^st ‘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)))) → (1^st ‘𝑔) ∈ ((1^st ‘𝑡)(Hom ‘𝐶)(1^st ‘𝑢)))
75	2, 11, 12, 34, 40, 38, 69, 74	catrid 17592	. . . . . 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)))) → ((1^st ‘𝑔)(⟨(1^st ‘𝑡), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(𝐼‘(1^st ‘𝑡))) = (1^st ‘𝑔))
76	66, 75	eqtrid 2780	. . . . 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)))) → ((1^st ‘𝑔)(⟨(1^st ‘𝑡), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)) = (1^st ‘𝑔))
77	48	oveq2i 7363	. . . . . 6 ⊢ ((2^nd ‘𝑔)(⟨(2^nd ‘𝑡), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)) = ((2^nd ‘𝑔)(⟨(2^nd ‘𝑡), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(𝐽‘(2^nd ‘𝑡)))
78		xp2nd 7960	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (2^nd ‘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)))) → (2^nd ‘𝑢) ∈ 𝑌)
80		xp2nd 7960	. . . . . . . 8 ⊢ (𝑔 ∈ (((1^st ‘𝑡)(Hom ‘𝐶)(1^st ‘𝑢)) × ((2^nd ‘𝑡)(Hom ‘𝐷)(2^nd ‘𝑢))) → (2^nd ‘𝑔) ∈ ((2^nd ‘𝑡)(Hom ‘𝐷)(2^nd ‘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)))) → (2^nd ‘𝑔) ∈ ((2^nd ‘𝑡)(Hom ‘𝐷)(2^nd ‘𝑢)))
82	3, 18, 19, 50, 54, 53, 79, 81	catrid 17592	. . . . . 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)))) → ((2^nd ‘𝑔)(⟨(2^nd ‘𝑡), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(𝐽‘(2^nd ‘𝑡))) = (2^nd ‘𝑔))
83	77, 82	eqtrid 2780	. . . . 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)))) → ((2^nd ‘𝑔)(⟨(2^nd ‘𝑡), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)) = (2^nd ‘𝑔))
84	76, 83	opeq12d 4832	. . . 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)))) → ⟨((1^st ‘𝑔)(⟨(1^st ‘𝑡), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)), ((2^nd ‘𝑔)(⟨(2^nd ‘𝑡), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩))⟩ = ⟨(1^st ‘𝑔), (2^nd ‘𝑔)⟩)
85	1, 4, 26, 38, 53, 60, 39, 39, 67, 61, 70	xpcco 18091	. . . 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‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩) = ⟨((1^st ‘𝑔)(⟨(1^st ‘𝑡), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)), ((2^nd ‘𝑔)(⟨(2^nd ‘𝑡), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩))⟩)
86		1st2nd2 7966	. . . . 5 ⊢ (𝑔 ∈ (((1^st ‘𝑡)(Hom ‘𝐶)(1^st ‘𝑢)) × ((2^nd ‘𝑡)(Hom ‘𝐷)(2^nd ‘𝑢))) → 𝑔 = ⟨(1^st ‘𝑔), (2^nd ‘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 = ⟨(1^st ‘𝑔), (2^nd ‘𝑔)⟩)
88	84, 85, 87	3eqtr4d 2778	. . 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‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩) = 𝑔)
89	2, 11, 38, 34, 37, 40, 69, 45, 74	catcocl 17593	. . . . 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)))) → ((1^st ‘𝑔)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘𝑓)) ∈ ((1^st ‘𝑠)(Hom ‘𝐶)(1^st ‘𝑢)))
90	3, 18, 53, 50, 52, 54, 79, 56, 81	catcocl 17593	. . . . 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)))) → ((2^nd ‘𝑔)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘𝑓)) ∈ ((2^nd ‘𝑠)(Hom ‘𝐷)(2^nd ‘𝑢)))
91	89, 90	opelxpd 5658	. . . 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)))) → ⟨((1^st ‘𝑔)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘𝑓))⟩ ∈ (((1^st ‘𝑠)(Hom ‘𝐶)(1^st ‘𝑢)) × ((2^nd ‘𝑠)(Hom ‘𝐷)(2^nd ‘𝑢))))
92	1, 4, 26, 38, 53, 60, 35, 39, 67, 41, 70	xpcco 18091	. . . 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) = ⟨((1^st ‘𝑔)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘𝑓))⟩)
93	1, 4, 11, 18, 26, 35, 67	xpchom 18088	. . . 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) = (((1^st ‘𝑠)(Hom ‘𝐶)(1^st ‘𝑢)) × ((2^nd ‘𝑠)(Hom ‘𝐷)(2^nd ‘𝑢))))
94	91, 92, 93	3eltr4d 2848	. . 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 1234	. . . . . . . 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 7959	. . . . . . . 8 ⊢ (𝑣 ∈ (𝑋 × 𝑌) → (1^st ‘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)))) → (1^st ‘𝑣) ∈ 𝑋)
98		simpr33 1266	. . . . . . . . 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 18088	. . . . . . . . 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) = (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))
100	98, 99	eleqtrd 2835	. . . . . . . 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 ∈ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))
101		xp1st 7959	. . . . . . . 8 ⊢ (ℎ ∈ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))) → (1^st ‘ℎ) ∈ ((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘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)))) → (1^st ‘ℎ) ∈ ((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)))
103	2, 11, 38, 34, 37, 40, 69, 45, 74, 97, 102	catass 17594	. . . . . 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)))) → (((1^st ‘ℎ)(⟨(1^st ‘𝑡), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑔))(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑓)) = ((1^st ‘ℎ)(⟨(1^st ‘𝑠), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))((1^st ‘𝑔)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘𝑓))))
104	1, 4, 26, 38, 53, 60, 39, 67, 95, 70, 98	xpcco 18091	. . . . . . . . 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) = ⟨((1^st ‘ℎ)(⟨(1^st ‘𝑡), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑔)), ((2^nd ‘ℎ)(⟨(2^nd ‘𝑡), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑔))⟩)
105	104	fveq2d 6832	. . . . . . . 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)))) → (1^st ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)) = (1^st ‘⟨((1^st ‘ℎ)(⟨(1^st ‘𝑡), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑔)), ((2^nd ‘ℎ)(⟨(2^nd ‘𝑡), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑔))⟩))
106		ovex 7385	. . . . . . . . 9 ⊢ ((1^st ‘ℎ)(⟨(1^st ‘𝑡), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑔)) ∈ V
107		ovex 7385	. . . . . . . . 9 ⊢ ((2^nd ‘ℎ)(⟨(2^nd ‘𝑡), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑔)) ∈ V
108	106, 107	op1st 7935	. . . . . . . 8 ⊢ (1^st ‘⟨((1^st ‘ℎ)(⟨(1^st ‘𝑡), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑔)), ((2^nd ‘ℎ)(⟨(2^nd ‘𝑡), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑔))⟩) = ((1^st ‘ℎ)(⟨(1^st ‘𝑡), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑔))
109	105, 108	eqtrdi 2784	. . . . . . 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)))) → (1^st ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)) = ((1^st ‘ℎ)(⟨(1^st ‘𝑡), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑔)))
110	109	oveq1d 7367	. . . . . 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)))) → ((1^st ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑓)) = (((1^st ‘ℎ)(⟨(1^st ‘𝑡), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑔))(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑓)))
111	92	fveq2d 6832	. . . . . . . 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)))) → (1^st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)) = (1^st ‘⟨((1^st ‘𝑔)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘𝑓))⟩))
112		ovex 7385	. . . . . . . . 9 ⊢ ((1^st ‘𝑔)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘𝑓)) ∈ V
113		ovex 7385	. . . . . . . . 9 ⊢ ((2^nd ‘𝑔)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘𝑓)) ∈ V
114	112, 113	op1st 7935	. . . . . . . 8 ⊢ (1^st ‘⟨((1^st ‘𝑔)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘𝑓))⟩) = ((1^st ‘𝑔)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘𝑓))
115	111, 114	eqtrdi 2784	. . . . . . 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)))) → (1^st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)) = ((1^st ‘𝑔)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘𝑓)))
116	115	oveq2d 7368	. . . . . 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)))) → ((1^st ‘ℎ)(⟨(1^st ‘𝑠), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))) = ((1^st ‘ℎ)(⟨(1^st ‘𝑠), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))((1^st ‘𝑔)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘𝑓))))
117	103, 110, 116	3eqtr4d 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)))) → ((1^st ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑓)) = ((1^st ‘ℎ)(⟨(1^st ‘𝑠), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))))
118		xp2nd 7960	. . . . . . . 8 ⊢ (𝑣 ∈ (𝑋 × 𝑌) → (2^nd ‘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)))) → (2^nd ‘𝑣) ∈ 𝑌)
120		xp2nd 7960	. . . . . . . 8 ⊢ (ℎ ∈ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))) → (2^nd ‘ℎ) ∈ ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘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)))) → (2^nd ‘ℎ) ∈ ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣)))
122	3, 18, 53, 50, 52, 54, 79, 56, 81, 119, 121	catass 17594	. . . . . 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)))) → (((2^nd ‘ℎ)(⟨(2^nd ‘𝑡), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑔))(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑓)) = ((2^nd ‘ℎ)(⟨(2^nd ‘𝑠), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))((2^nd ‘𝑔)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘𝑓))))
123	104	fveq2d 6832	. . . . . . . 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)))) → (2^nd ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)) = (2^nd ‘⟨((1^st ‘ℎ)(⟨(1^st ‘𝑡), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑔)), ((2^nd ‘ℎ)(⟨(2^nd ‘𝑡), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑔))⟩))
124	106, 107	op2nd 7936	. . . . . . . 8 ⊢ (2^nd ‘⟨((1^st ‘ℎ)(⟨(1^st ‘𝑡), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑔)), ((2^nd ‘ℎ)(⟨(2^nd ‘𝑡), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑔))⟩) = ((2^nd ‘ℎ)(⟨(2^nd ‘𝑡), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑔))
125	123, 124	eqtrdi 2784	. . . . . . 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)))) → (2^nd ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)) = ((2^nd ‘ℎ)(⟨(2^nd ‘𝑡), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑔)))
126	125	oveq1d 7367	. . . . . 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)))) → ((2^nd ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑓)) = (((2^nd ‘ℎ)(⟨(2^nd ‘𝑡), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑔))(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑓)))
127	92	fveq2d 6832	. . . . . . . 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)))) → (2^nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)) = (2^nd ‘⟨((1^st ‘𝑔)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘𝑓))⟩))
128	112, 113	op2nd 7936	. . . . . . . 8 ⊢ (2^nd ‘⟨((1^st ‘𝑔)(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑢))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘𝑓))⟩) = ((2^nd ‘𝑔)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘𝑓))
129	127, 128	eqtrdi 2784	. . . . . . 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)))) → (2^nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)) = ((2^nd ‘𝑔)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘𝑓)))
130	129	oveq2d 7368	. . . . . 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)))) → ((2^nd ‘ℎ)(⟨(2^nd ‘𝑠), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))) = ((2^nd ‘ℎ)(⟨(2^nd ‘𝑠), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))((2^nd ‘𝑔)(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑢))(2^nd ‘𝑓))))
131	122, 126, 130	3eqtr4d 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)))) → ((2^nd ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑓)) = ((2^nd ‘ℎ)(⟨(2^nd ‘𝑠), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))))
132	117, 131	opeq12d 4832	. . . 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)))) → ⟨((1^st ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑓)), ((2^nd ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑓))⟩ = ⟨((1^st ‘ℎ)(⟨(1^st ‘𝑠), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))), ((2^nd ‘ℎ)(⟨(2^nd ‘𝑠), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)))⟩)
133	2, 11, 38, 34, 40, 69, 97, 74, 102	catcocl 17593	. . . . . . 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)))) → ((1^st ‘ℎ)(⟨(1^st ‘𝑡), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑔)) ∈ ((1^st ‘𝑡)(Hom ‘𝐶)(1^st ‘𝑣)))
134	3, 18, 53, 50, 54, 79, 119, 81, 121	catcocl 17593	. . . . . . 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)))) → ((2^nd ‘ℎ)(⟨(2^nd ‘𝑡), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑔)) ∈ ((2^nd ‘𝑡)(Hom ‘𝐷)(2^nd ‘𝑣)))
135	133, 134	opelxpd 5658	. . . . . 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)))) → ⟨((1^st ‘ℎ)(⟨(1^st ‘𝑡), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑔)), ((2^nd ‘ℎ)(⟨(2^nd ‘𝑡), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑔))⟩ ∈ (((1^st ‘𝑡)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑡)(Hom ‘𝐷)(2^nd ‘𝑣))))
136	1, 4, 11, 18, 26, 39, 95	xpchom 18088	. . . . . 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) = (((1^st ‘𝑡)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑡)(Hom ‘𝐷)(2^nd ‘𝑣))))
137	135, 104, 136	3eltr4d 2848	. . . . 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 18091	. . . 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) = ⟨((1^st ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(1^st ‘𝑠), (1^st ‘𝑡)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘𝑓)), ((2^nd ‘(ℎ(⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(2^nd ‘𝑠), (2^nd ‘𝑡)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘𝑓))⟩)
139	1, 4, 26, 38, 53, 60, 35, 67, 95, 94, 98	xpcco 18091	. . . 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)) = ⟨((1^st ‘ℎ)(⟨(1^st ‘𝑠), (1^st ‘𝑢)⟩(comp‘𝐶)(1^st ‘𝑣))(1^st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))), ((2^nd ‘ℎ)(⟨(2^nd ‘𝑠), (2^nd ‘𝑢)⟩(comp‘𝐷)(2^nd ‘𝑣))(2^nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)))⟩)
140	132, 138, 139	3eqtr4d 2778	. . 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 17589	. 2 ⊢ (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑡 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)))
142		vex 3441	. . . . . . . 8 ⊢ 𝑥 ∈ V
143		vex 3441	. . . . . . . 8 ⊢ 𝑦 ∈ V
144	142, 143	op1std 7937	. . . . . . 7 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑡) = 𝑥)
145	144	fveq2d 6832	. . . . . 6 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝐼‘(1^st ‘𝑡)) = (𝐼‘𝑥))
146	142, 143	op2ndd 7938	. . . . . . 7 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝑡) = 𝑦)
147	146	fveq2d 6832	. . . . . 6 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝐽‘(2^nd ‘𝑡)) = (𝐽‘𝑦))
148	145, 147	opeq12d 4832	. . . . 5 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → ⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩ = ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)
149	148	mpompt 7466	. . . 4 ⊢ (𝑡 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)
150	149	eqeq2i 2746	. . 3 ⊢ ((Id‘𝑇) = (𝑡 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩) ↔ (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩))
151	150	anbi2i 623	. 2 ⊢ ((𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑡 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐼‘(1^st ‘𝑡)), (𝐽‘(2^nd ‘𝑡))⟩)) ↔ (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)))
152	141, 151	sylib 218	1 ⊢ (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 Vcvv 3437 ⟨cop 4581 ↦ cmpt 5174 × cxp 5617 ‘cfv 6486 (class class class)co 7352 ∈ cmpo 7354 1^st c1st 7925 2^nd c2nd 7926 Basecbs 17122 Hom chom 17174 compcco 17175 Catccat 17572 Idccid 17573 ×_c cxpc 18076

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-fz 13410 df-struct 17060 df-slot 17095 df-ndx 17107 df-base 17123 df-hom 17187 df-cco 17188 df-cat 17576 df-cid 17577 df-xpc 18080

This theorem is referenced by: xpcid 18097 xpccat 18098

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