Theorem cofucl 18063

Description: The composition of two functors is a functor. Proposition 3.23 of [Adamek] p. 33. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
cofucl.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
cofucl.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))

Assertion

Ref	Expression
cofucl	⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Func 𝐸))

Proof of Theorem cofucl

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

Step	Hyp	Ref	Expression
1		eqid 2761	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		cofucl.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		cofucl.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
4	1, 2, 3	cofuval 18057	. . 3 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = ⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩)
5	1, 2, 3	cofu1st 18058	. . . 4 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = ((1st ‘𝐺) ∘ (1st ‘𝐹)))
6	4	fveq2d 6889	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐹)) = (2nd ‘⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩))
7		fvex 6898	. . . . . . 7 ⊢ (1st ‘𝐺) ∈ V
8		fvex 6898	. . . . . . 7 ⊢ (1st ‘𝐹) ∈ V
9	7, 8	coex 7942	. . . . . 6 ⊢ ((1st ‘𝐺) ∘ (1st ‘𝐹)) ∈ V
10		fvex 6898	. . . . . . 7 ⊢ (Base‘𝐶) ∈ V
11	10, 10	mpoex 8092	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) ∈ V
12	9, 11	op2nd 8010	. . . . 5 ⊢ (2nd ‘⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))
13	6, 12	eqtrdi 2812	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐹)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))))
14	5, 13	opeq12d 4841	. . 3 ⊢ (𝜑 → ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩ = ⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩)
15	4, 14	eqtr4d 2799	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩)
16		eqid 2761	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
17		eqid 2761	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
18		relfunc 18037	. . . . . . . 8 ⊢ Rel (𝐷 Func 𝐸)
19		1st2ndbr 8053	. . . . . . . 8 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐺 ∈ (𝐷 Func 𝐸)) → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
20	18, 3, 19	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
21	16, 17, 20	funcf1 18041	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐺):(Base‘𝐷)⟶(Base‘𝐸))
22		relfunc 18037	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
23		1st2ndbr 8053	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
24	22, 2, 23	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
25	1, 16, 24	funcf1 18041	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
26		fco 6734	. . . . . 6 ⊢ (((1st ‘𝐺):(Base‘𝐷)⟶(Base‘𝐸) ∧ (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷)) → ((1st ‘𝐺) ∘ (1st ‘𝐹)):(Base‘𝐶)⟶(Base‘𝐸))
27	21, 25, 26	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐺) ∘ (1st ‘𝐹)):(Base‘𝐶)⟶(Base‘𝐸))
28	5	feq1d 6691	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹)):(Base‘𝐶)⟶(Base‘𝐸) ↔ ((1st ‘𝐺) ∘ (1st ‘𝐹)):(Base‘𝐶)⟶(Base‘𝐸)))
29	27, 28	mpbird 260	. . . 4 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)):(Base‘𝐶)⟶(Base‘𝐸))
30		eqid 2761	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))
31		ovex 7453	. . . . . . . 8 ⊢ (((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∈ V
32		ovex 7453	. . . . . . . 8 ⊢ (𝑥(2nd ‘𝐹)𝑦) ∈ V
33	31, 32	coex 7942	. . . . . . 7 ⊢ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)) ∈ V
34	30, 33	fnmpoi 8081	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) Fn ((Base‘𝐶) × (Base‘𝐶))
35	13	fneq1d 6632	. . . . . 6 ⊢ (𝜑 → ((2nd ‘(𝐺 ∘func 𝐹)) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) Fn ((Base‘𝐶) × (Base‘𝐶))))
36	34, 35	mpbiri 261	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐹)) Fn ((Base‘𝐶) × (Base‘𝐶)))
37		eqid 2761	. . . . . . . . . . 11 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
38		eqid 2761	. . . . . . . . . . 11 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
39	20	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
40	25	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
41		simprl 783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
42	40, 41	ffvelcdmd 7085	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
43		simprr 785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
44	40, 43	ffvelcdmd 7085	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
45	16, 37, 38, 39, 42, 44	funcf2 18043	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)):(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))⟶(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))))
46		eqid 2761	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
47	24	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
48	1, 46, 37, 47, 41, 43	funcf2 18043	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
49		fco 6734	. . . . . . . . . 10 ⊢ (((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)):(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))⟶(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))) ∧ (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))))
50	45, 48, 49	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))))
51		ovex 7453	. . . . . . . . . 10 ⊢ (((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))) ∈ V
52		ovex 7453	. . . . . . . . . 10 ⊢ (𝑥(Hom ‘𝐶)𝑦) ∈ V
53	51, 52	elmap 8899	. . . . . . . . 9 ⊢ (((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)) ∈ ((((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))) ↑m (𝑥(Hom ‘𝐶)𝑦)) ↔ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))))
54	50, 53	sylibr 237	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)) ∈ ((((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))) ↑m (𝑥(Hom ‘𝐶)𝑦)))
55	2	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Func 𝐷))
56	3	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Func 𝐸))
57	1, 55, 56, 41, 43	cofu2nd 18060	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦) = ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))
58	1, 55, 56, 41	cofu1 18059	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑥) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)))
59	1, 55, 56, 43	cofu1 18059	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑦) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦)))
60	58, 59	oveq12d 7438	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦)) = (((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))))
61	60	oveq1d 7435	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦)) ↑m (𝑥(Hom ‘𝐶)𝑦)) = ((((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))) ↑m (𝑥(Hom ‘𝐶)𝑦)))
62	54, 57, 61	3eltr4d 2876	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦) ∈ ((((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦)) ↑m (𝑥(Hom ‘𝐶)𝑦)))
63	62	ralrimivva 3206	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦) ∈ ((((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦)) ↑m (𝑥(Hom ‘𝐶)𝑦)))
64		fveq2 6885	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd ‘(𝐺 ∘func 𝐹))‘𝑧) = ((2nd ‘(𝐺 ∘func 𝐹))‘⟨𝑥, 𝑦⟩))
65		df-ov 7423	. . . . . . . . 9 ⊢ (𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦) = ((2nd ‘(𝐺 ∘func 𝐹))‘⟨𝑥, 𝑦⟩)
66	64, 65	eqtr4di 2814	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd ‘(𝐺 ∘func 𝐹))‘𝑧) = (𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦))
67		vex 3455	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
68		vex 3455	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
69	67, 68	op1std 8011	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
70	69	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st ‘(𝐺 ∘func 𝐹))‘(1st ‘𝑧)) = ((1st ‘(𝐺 ∘func 𝐹))‘𝑥))
71	67, 68	op2ndd 8012	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
72	71	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st ‘(𝐺 ∘func 𝐹))‘(2nd ‘𝑧)) = ((1st ‘(𝐺 ∘func 𝐹))‘𝑦))
73	70, 72	oveq12d 7438	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((1st ‘(𝐺 ∘func 𝐹))‘(1st ‘𝑧))(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘(2nd ‘𝑧))) = (((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦)))
74		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((Hom ‘𝐶)‘𝑧) = ((Hom ‘𝐶)‘⟨𝑥, 𝑦⟩))
75		df-ov 7423	. . . . . . . . . 10 ⊢ (𝑥(Hom ‘𝐶)𝑦) = ((Hom ‘𝐶)‘⟨𝑥, 𝑦⟩)
76	74, 75	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((Hom ‘𝐶)‘𝑧) = (𝑥(Hom ‘𝐶)𝑦))
77	73, 76	oveq12d 7438	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((((1st ‘(𝐺 ∘func 𝐹))‘(1st ‘𝑧))(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) = ((((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦)) ↑m (𝑥(Hom ‘𝐶)𝑦)))
78	66, 77	eleq12d 2855	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((2nd ‘(𝐺 ∘func 𝐹))‘𝑧) ∈ ((((1st ‘(𝐺 ∘func 𝐹))‘(1st ‘𝑧))(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ↔ (𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦) ∈ ((((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦)) ↑m (𝑥(Hom ‘𝐶)𝑦))))
79	78	ralxp 5818	. . . . . 6 ⊢ (∀𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))((2nd ‘(𝐺 ∘func 𝐹))‘𝑧) ∈ ((((1st ‘(𝐺 ∘func 𝐹))‘(1st ‘𝑧))(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦) ∈ ((((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦)) ↑m (𝑥(Hom ‘𝐶)𝑦)))
80	63, 79	sylibr 237	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))((2nd ‘(𝐺 ∘func 𝐹))‘𝑧) ∈ ((((1st ‘(𝐺 ∘func 𝐹))‘(1st ‘𝑧))(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)))
81		fvex 6898	. . . . . 6 ⊢ (2nd ‘(𝐺 ∘func 𝐹)) ∈ V
82	81	elixp 8932	. . . . 5 ⊢ ((2nd ‘(𝐺 ∘func 𝐹)) ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))((((1st ‘(𝐺 ∘func 𝐹))‘(1st ‘𝑧))(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ↔ ((2nd ‘(𝐺 ∘func 𝐹)) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ∀𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))((2nd ‘(𝐺 ∘func 𝐹))‘𝑧) ∈ ((((1st ‘(𝐺 ∘func 𝐹))‘(1st ‘𝑧))(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧))))
83	36, 80, 82	sylanbrc 595	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐹)) ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))((((1st ‘(𝐺 ∘func 𝐹))‘(1st ‘𝑧))(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)))
84		eqid 2761	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
85		eqid 2761	. . . . . . . . . 10 ⊢ (Id‘𝐷) = (Id‘𝐷)
86	24	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
87		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
88	1, 84, 85, 86, 87	funcid 18045	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd ‘𝐹)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘((1st ‘𝐹)‘𝑥)))
89	88	fveq2d 6889	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑥))‘((𝑥(2nd ‘𝐹)𝑥)‘((Id‘𝐶)‘𝑥))) = ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑥))‘((Id‘𝐷)‘((1st ‘𝐹)‘𝑥))))
90		eqid 2761	. . . . . . . . 9 ⊢ (Id‘𝐸) = (Id‘𝐸)
91	20	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
92	25	ffvelcdmda 7084	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
93	16, 85, 90, 91, 92	funcid 18045	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑥))‘((Id‘𝐷)‘((1st ‘𝐹)‘𝑥))) = ((Id‘𝐸)‘((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))))
94	89, 93	eqtrd 2796	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑥))‘((𝑥(2nd ‘𝐹)𝑥)‘((Id‘𝐶)‘𝑥))) = ((Id‘𝐸)‘((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))))
95	2	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ (𝐶 Func 𝐷))
96	3	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐺 ∈ (𝐷 Func 𝐸))
97		funcrcl 18038	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
98	2, 97	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
99	98	simpld 500	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ Cat)
100	99	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
101	1, 46, 84, 100, 87	catidcl 17856	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
102	1, 95, 96, 87, 87, 46, 101	cofu2 18061	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑥)‘((Id‘𝐶)‘𝑥)) = ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑥))‘((𝑥(2nd ‘𝐹)𝑥)‘((Id‘𝐶)‘𝑥))))
103	1, 95, 96, 87	cofu1 18059	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑥) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)))
104	103	fveq2d 6889	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐸)‘((1st ‘(𝐺 ∘func 𝐹))‘𝑥)) = ((Id‘𝐸)‘((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))))
105	94, 102, 104	3eqtr4d 2806	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐸)‘((1st ‘(𝐺 ∘func 𝐹))‘𝑥)))
106	86	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
107		simplr 781	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥 ∈ (Base‘𝐶))
108		simprlr 792	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧 ∈ (Base‘𝐶))
109	1, 46, 37, 106, 107, 108	funcf2 18043	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑥(2nd ‘𝐹)𝑧):(𝑥(Hom ‘𝐶)𝑧)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑧)))
110		eqid 2761	. . . . . . . . . . . . 13 ⊢ (comp‘𝐶) = (comp‘𝐶)
111	100	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝐶 ∈ Cat)
112		simprll 791	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑦 ∈ (Base‘𝐶))
113		simprrl 793	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
114		simprrr 794	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
115	1, 46, 110, 111, 107, 112, 108, 113, 114	catcocl 17859	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
116		fvco3 6985	. . . . . . . . . . . 12 ⊢ (((𝑥(2nd ‘𝐹)𝑧):(𝑥(Hom ‘𝐶)𝑧)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) → (((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑧)) ∘ (𝑥(2nd ‘𝐹)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑧))‘((𝑥(2nd ‘𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))
117	109, 115, 116	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑧)) ∘ (𝑥(2nd ‘𝐹)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑧))‘((𝑥(2nd ‘𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))
118		eqid 2761	. . . . . . . . . . . . 13 ⊢ (comp‘𝐷) = (comp‘𝐷)
119	1, 46, 110, 118, 106, 107, 112, 108, 113, 114	funcco 18046	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑥(2nd ‘𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd ‘𝐹)𝑧)‘𝑔)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐹)‘𝑧))((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
120	119	fveq2d 6889	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑧))‘((𝑥(2nd ‘𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) = ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑧))‘(((𝑦(2nd ‘𝐹)𝑧)‘𝑔)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐹)‘𝑧))((𝑥(2nd ‘𝐹)𝑦)‘𝑓))))
121		eqid 2761	. . . . . . . . . . . 12 ⊢ (comp‘𝐸) = (comp‘𝐸)
122	91	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
123	92	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
124	25	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
125	124	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
126	125, 112	ffvelcdmd 7085	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
127	125, 108	ffvelcdmd 7085	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((1st ‘𝐹)‘𝑧) ∈ (Base‘𝐷))
128	1, 46, 37, 106, 107, 112	funcf2 18043	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
129	128, 113	ffvelcdmd 7085	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑥(2nd ‘𝐹)𝑦)‘𝑓) ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
130	1, 46, 37, 106, 112, 108	funcf2 18043	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑦(2nd ‘𝐹)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st ‘𝐹)‘𝑦)(Hom ‘𝐷)((1st ‘𝐹)‘𝑧)))
131	130, 114	ffvelcdmd 7085	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑦(2nd ‘𝐹)𝑧)‘𝑔) ∈ (((1st ‘𝐹)‘𝑦)(Hom ‘𝐷)((1st ‘𝐹)‘𝑧)))
132	16, 37, 118, 121, 122, 123, 126, 127, 129, 131	funcco 18046	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑧))‘(((𝑦(2nd ‘𝐹)𝑧)‘𝑔)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐹)‘𝑧))((𝑥(2nd ‘𝐹)𝑦)‘𝑓))) = (((((1st ‘𝐹)‘𝑦)(2nd ‘𝐺)((1st ‘𝐹)‘𝑧))‘((𝑦(2nd ‘𝐹)𝑧)‘𝑔))(⟨((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)), ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))⟩(comp‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑧)))((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦))‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓))))
133	117, 120, 132	3eqtrd 2800	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑧)) ∘ (𝑥(2nd ‘𝐹)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((((1st ‘𝐹)‘𝑦)(2nd ‘𝐺)((1st ‘𝐹)‘𝑧))‘((𝑦(2nd ‘𝐹)𝑧)‘𝑔))(⟨((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)), ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))⟩(comp‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑧)))((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦))‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓))))
134	95	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝐹 ∈ (𝐶 Func 𝐷))
135	96	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝐺 ∈ (𝐷 Func 𝐸))
136	1, 134, 135, 107, 108	cofu2nd 18060	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑧) = ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑧)) ∘ (𝑥(2nd ‘𝐹)𝑧)))
137	136	fveq1d 6887	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑧)) ∘ (𝑥(2nd ‘𝐹)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))
138	103	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑥) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)))
139	1, 134, 135, 112	cofu1 18059	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑦) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦)))
140	138, 139	opeq12d 4841	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ⟨((1st ‘(𝐺 ∘func 𝐹))‘𝑥), ((1st ‘(𝐺 ∘func 𝐹))‘𝑦)⟩ = ⟨((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)), ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))⟩)
141	1, 134, 135, 108	cofu1 18059	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑧) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑧)))
142	140, 141	oveq12d 7438	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (⟨((1st ‘(𝐺 ∘func 𝐹))‘𝑥), ((1st ‘(𝐺 ∘func 𝐹))‘𝑦)⟩(comp‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑧)) = (⟨((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)), ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))⟩(comp‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑧))))
143	1, 134, 135, 112, 108, 46, 114	cofu2 18061	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑦(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘𝑔) = ((((1st ‘𝐹)‘𝑦)(2nd ‘𝐺)((1st ‘𝐹)‘𝑧))‘((𝑦(2nd ‘𝐹)𝑧)‘𝑔)))
144	1, 134, 135, 107, 112, 46, 113	cofu2 18061	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦)‘𝑓) = ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦))‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
145	142, 143, 144	oveq123d 7441	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (((𝑦(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘𝑔)(⟨((1st ‘(𝐺 ∘func 𝐹))‘𝑥), ((1st ‘(𝐺 ∘func 𝐹))‘𝑦)⟩(comp‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑧))((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦)‘𝑓)) = (((((1st ‘𝐹)‘𝑦)(2nd ‘𝐺)((1st ‘𝐹)‘𝑧))‘((𝑦(2nd ‘𝐹)𝑧)‘𝑔))(⟨((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)), ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))⟩(comp‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑧)))((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦))‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓))))
146	133, 137, 145	3eqtr4d 2806	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘𝑔)(⟨((1st ‘(𝐺 ∘func 𝐹))‘𝑥), ((1st ‘(𝐺 ∘func 𝐹))‘𝑦)⟩(comp‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑧))((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦)‘𝑓)))
147	146	anassrs 473	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘𝑔)(⟨((1st ‘(𝐺 ∘func 𝐹))‘𝑥), ((1st ‘(𝐺 ∘func 𝐹))‘𝑦)⟩(comp‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑧))((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦)‘𝑓)))
148	147	ralrimivva 3206	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘𝑔)(⟨((1st ‘(𝐺 ∘func 𝐹))‘𝑥), ((1st ‘(𝐺 ∘func 𝐹))‘𝑦)⟩(comp‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑧))((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦)‘𝑓)))
149	148	ralrimivva 3206	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘𝑔)(⟨((1st ‘(𝐺 ∘func 𝐹))‘𝑥), ((1st ‘(𝐺 ∘func 𝐹))‘𝑦)⟩(comp‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑧))((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦)‘𝑓)))
150	105, 149	jca 521	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐸)‘((1st ‘(𝐺 ∘func 𝐹))‘𝑥)) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘𝑔)(⟨((1st ‘(𝐺 ∘func 𝐹))‘𝑥), ((1st ‘(𝐺 ∘func 𝐹))‘𝑦)⟩(comp‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑧))((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦)‘𝑓))))
151	150	ralrimiva 3155	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)(((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐸)‘((1st ‘(𝐺 ∘func 𝐹))‘𝑥)) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘𝑔)(⟨((1st ‘(𝐺 ∘func 𝐹))‘𝑥), ((1st ‘(𝐺 ∘func 𝐹))‘𝑦)⟩(comp‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑧))((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦)‘𝑓))))
152		funcrcl 18038	. . . . . . 7 ⊢ (𝐺 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
153	3, 152	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
154	153	simprd 501	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
155	1, 17, 46, 38, 84, 90, 110, 121, 99, 154	isfunc 18039	. . . 4 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))(𝐶 Func 𝐸)(2nd ‘(𝐺 ∘func 𝐹)) ↔ ((1st ‘(𝐺 ∘func 𝐹)):(Base‘𝐶)⟶(Base‘𝐸) ∧ (2nd ‘(𝐺 ∘func 𝐹)) ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))((((1st ‘(𝐺 ∘func 𝐹))‘(1st ‘𝑧))(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ∧ ∀𝑥 ∈ (Base‘𝐶)(((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐸)‘((1st ‘(𝐺 ∘func 𝐹))‘𝑥)) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd ‘(𝐺 ∘func 𝐹))𝑧)‘𝑔)(⟨((1st ‘(𝐺 ∘func 𝐹))‘𝑥), ((1st ‘(𝐺 ∘func 𝐹))‘𝑦)⟩(comp‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑧))((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦)‘𝑓))))))
156	29, 83, 151, 155	mpbir3and 1361	. . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹))(𝐶 Func 𝐸)(2nd ‘(𝐺 ∘func 𝐹)))
157		df-br 5104	. . 3 ⊢ ((1st ‘(𝐺 ∘func 𝐹))(𝐶 Func 𝐸)(2nd ‘(𝐺 ∘func 𝐹)) ↔ ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩ ∈ (𝐶 Func 𝐸))
158	156, 157	sylib 221	. 2 ⊢ (𝜑 → ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩ ∈ (𝐶 Func 𝐸))
159	15, 158	eqeltrd 2861	1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Func 𝐸))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ⟨cop 4590   class class class wbr 5103   × cxp 5649   ∘ ccom 5655  Rel wrel 5656   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000   ↑_m cmap 8847  Xcixp 8925  Basecbs 17387  Hom chom 17439  compcco 17440  Catccat 17838  Idccid 17839   Func cfunc 18029   ∘_func ccofu 18031

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8849  df-ixp 8926  df-cat 17842  df-cid 17843  df-func 18033  df-cofu 18035

This theorem is used by:  cofuass  18064  cofull  18111  cofth  18112  catccatid  18281  1st2ndprf  18380  uncfcl  18409  uncf1  18410  uncf2  18411  yonedalem1  18446  yonedalem21  18447  yonedalem22  18452  funcrngcsetcALT  20893  rescofuf  50200  cofu1a  50201  cofu2a  50202  cofucla  50203  cofuoppf  50257  uptrlem2  50318  uptra  50322  uptr2a  50329  cofuswapfcl  50400  prcofdiag1  50500  prcofdiag  50501  oppfdiag1  50521  oppfdiag  50523  cofuterm  50652