Theorem cofucl 17810

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 2734	. . . 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 17804	. . 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 17805	. . . 4 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = ((1st ‘𝐺) ∘ (1st ‘𝐹)))
6	4	fveq2d 6836	. . . . 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 6845	. . . . . . 7 ⊢ (1st ‘𝐺) ∈ V
8		fvex 6845	. . . . . . 7 ⊢ (1st ‘𝐹) ∈ V
9	7, 8	coex 7870	. . . . . 6 ⊢ ((1st ‘𝐺) ∘ (1st ‘𝐹)) ∈ V
10		fvex 6845	. . . . . . 7 ⊢ (Base‘𝐶) ∈ V
11	10, 10	mpoex 8021	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) ∈ V
12	9, 11	op2nd 7940	. . . . 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 2785	. . . 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 4835	. . 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 2772	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩)
16		eqid 2734	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
17		eqid 2734	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
18		relfunc 17784	. . . . . . . 8 ⊢ Rel (𝐷 Func 𝐸)
19		1st2ndbr 7984	. . . . . . . 8 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐺 ∈ (𝐷 Func 𝐸)) → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
20	18, 3, 19	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
21	16, 17, 20	funcf1 17788	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐺):(Base‘𝐷)⟶(Base‘𝐸))
22		relfunc 17784	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
23		1st2ndbr 7984	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
24	22, 2, 23	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
25	1, 16, 24	funcf1 17788	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
26		fco 6684	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐺) ∘ (1st ‘𝐹)):(Base‘𝐶)⟶(Base‘𝐸))
28	5	feq1d 6642	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹)):(Base‘𝐶)⟶(Base‘𝐸) ↔ ((1st ‘𝐺) ∘ (1st ‘𝐹)):(Base‘𝐶)⟶(Base‘𝐸)))
29	27, 28	mpbird 257	. . . 4 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)):(Base‘𝐶)⟶(Base‘𝐸))
30		eqid 2734	. . . . . . 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 7389	. . . . . . . 8 ⊢ (((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∈ V
32		ovex 7389	. . . . . . . 8 ⊢ (𝑥(2nd ‘𝐹)𝑦) ∈ V
33	31, 32	coex 7870	. . . . . . 7 ⊢ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)) ∈ V
34	30, 33	fnmpoi 8012	. . . . . 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 6583	. . . . . 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 258	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐹)) Fn ((Base‘𝐶) × (Base‘𝐶)))
37		eqid 2734	. . . . . . . . . . 11 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
38		eqid 2734	. . . . . . . . . . 11 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
39	20	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
40	25	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
41		simprl 770	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
42	40, 41	ffvelcdmd 7028	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
43		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
44	40, 43	ffvelcdmd 7028	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
45	16, 37, 38, 39, 42, 44	funcf2 17790	. . . . . . . . . 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 2734	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
47	24	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
48	1, 46, 37, 47, 41, 43	funcf2 17790	. . . . . . . . . 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 6684	. . . . . . . . . 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 584	. . . . . . . . 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 7389	. . . . . . . . . 10 ⊢ (((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))) ∈ V
52		ovex 7389	. . . . . . . . . 10 ⊢ (𝑥(Hom ‘𝐶)𝑦) ∈ V
53	51, 52	elmap 8807	. . . . . . . . 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 234	. . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Func 𝐷))
56	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Func 𝐸))
57	1, 55, 56, 41, 43	cofu2nd 17807	. . . . . . . 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 17806	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑥) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)))
59	1, 55, 56, 43	cofu1 17806	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑦) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦)))
60	58, 59	oveq12d 7374	. . . . . . . . 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 7371	. . . . . . . 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 2849	. . . . . . 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 3177	. . . . . 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 6832	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd ‘(𝐺 ∘func 𝐹))‘𝑧) = ((2nd ‘(𝐺 ∘func 𝐹))‘⟨𝑥, 𝑦⟩))
65		df-ov 7359	. . . . . . . . 9 ⊢ (𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦) = ((2nd ‘(𝐺 ∘func 𝐹))‘⟨𝑥, 𝑦⟩)
66	64, 65	eqtr4di 2787	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd ‘(𝐺 ∘func 𝐹))‘𝑧) = (𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦))
67		vex 3442	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
68		vex 3442	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
69	67, 68	op1std 7941	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
70	69	fveq2d 6836	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st ‘(𝐺 ∘func 𝐹))‘(1st ‘𝑧)) = ((1st ‘(𝐺 ∘func 𝐹))‘𝑥))
71	67, 68	op2ndd 7942	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
72	71	fveq2d 6836	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st ‘(𝐺 ∘func 𝐹))‘(2nd ‘𝑧)) = ((1st ‘(𝐺 ∘func 𝐹))‘𝑦))
73	70, 72	oveq12d 7374	. . . . . . . . 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 6832	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((Hom ‘𝐶)‘𝑧) = ((Hom ‘𝐶)‘⟨𝑥, 𝑦⟩))
75		df-ov 7359	. . . . . . . . . 10 ⊢ (𝑥(Hom ‘𝐶)𝑦) = ((Hom ‘𝐶)‘⟨𝑥, 𝑦⟩)
76	74, 75	eqtr4di 2787	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((Hom ‘𝐶)‘𝑧) = (𝑥(Hom ‘𝐶)𝑦))
77	73, 76	oveq12d 7374	. . . . . . . 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 2828	. . . . . . 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 5788	. . . . . 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 234	. . . . 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 6845	. . . . . 6 ⊢ (2nd ‘(𝐺 ∘func 𝐹)) ∈ V
82	81	elixp 8840	. . . . 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 583	. . . 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 2734	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
85		eqid 2734	. . . . . . . . . 10 ⊢ (Id‘𝐷) = (Id‘𝐷)
86	24	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
87		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
88	1, 84, 85, 86, 87	funcid 17792	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd ‘𝐹)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘((1st ‘𝐹)‘𝑥)))
89	88	fveq2d 6836	. . . . . . . 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 2734	. . . . . . . . 9 ⊢ (Id‘𝐸) = (Id‘𝐸)
91	20	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
92	25	ffvelcdmda 7027	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
93	16, 85, 90, 91, 92	funcid 17792	. . . . . . . 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 2769	. . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ (𝐶 Func 𝐷))
96	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐺 ∈ (𝐷 Func 𝐸))
97		funcrcl 17785	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
98	2, 97	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
99	98	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ Cat)
100	99	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
101	1, 46, 84, 100, 87	catidcl 17603	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
102	1, 95, 96, 87, 87, 46, 101	cofu2 17808	. . . . . . 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 17806	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑥) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)))
104	103	fveq2d 6836	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐸)‘((1st ‘(𝐺 ∘func 𝐹))‘𝑥)) = ((Id‘𝐸)‘((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))))
105	94, 102, 104	3eqtr4d 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐸)‘((1st ‘(𝐺 ∘func 𝐹))‘𝑥)))
106	86	adantr 480	. . . . . . . . . . . . 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 768	. . . . . . . . . . . . 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 779	. . . . . . . . . . . . 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 17790	. . . . . . . . . . . 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 2734	. . . . . . . . . . . . 13 ⊢ (comp‘𝐶) = (comp‘𝐶)
111	100	adantr 480	. . . . . . . . . . . . 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 778	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . 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 17606	. . . . . . . . . . . 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 6931	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 2734	. . . . . . . . . . . . 13 ⊢ (comp‘𝐷) = (comp‘𝐷)
119	1, 46, 110, 118, 106, 107, 112, 108, 113, 114	funcco 17793	. . . . . . . . . . . 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 6836	. . . . . . . . . . 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 2734	. . . . . . . . . . . 12 ⊢ (comp‘𝐸) = (comp‘𝐸)
122	91	adantr 480	. . . . . . . . . . . 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 480	. . . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
125	124	adantr 480	. . . . . . . . . . . . 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 7028	. . . . . . . . . . . 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 7028	. . . . . . . . . . . 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 17790	. . . . . . . . . . . . 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 7028	. . . . . . . . . . . 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 17790	. . . . . . . . . . . . 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 7028	. . . . . . . . . . . 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 17793	. . . . . . . . . . 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 2773	. . . . . . . . . 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 480	. . . . . . . . . . . 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 480	. . . . . . . . . . . 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 17807	. . . . . . . . . . 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 6834	. . . . . . . . . 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 480	. . . . . . . . . . . . 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 17806	. . . . . . . . . . . . 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 4835	. . . . . . . . . . . 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 17806	. . . . . . . . . . . 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 7374	. . . . . . . . . . 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 17808	. . . . . . . . . . 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 17808	. . . . . . . . . . 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 7377	. . . . . . . . . 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 2779	. . . . . . . . 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 467	. . . . . . . 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 3177	. . . . . . 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 3177	. . . . . 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 511	. . . . 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 3126	. . . 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 17785	. . . . . . 7 ⊢ (𝐺 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
153	3, 152	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
154	153	simprd 495	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
155	1, 17, 46, 38, 84, 90, 110, 121, 99, 154	isfunc 17786	. . . 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 1343	. . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹))(𝐶 Func 𝐸)(2nd ‘(𝐺 ∘func 𝐹)))
157		df-br 5097	. . 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 218	. 2 ⊢ (𝜑 → ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩ ∈ (𝐶 Func 𝐸))
159	15, 158	eqeltrd 2834	1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Func 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ⟨cop 4584   class class class wbr 5096   × cxp 5620   ∘ ccom 5626  Rel wrel 5627   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930   ↑_m cmap 8761  Xcixp 8833  Basecbs 17134  Hom chom 17186  compcco 17187  Catccat 17585  Idccid 17586   Func cfunc 17776   ∘_func ccofu 17778

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 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8763  df-ixp 8834  df-cat 17589  df-cid 17590  df-func 17780  df-cofu 17782

This theorem is referenced by:  cofuass  17811  cofull  17858  cofth  17859  catccatid  18028  1st2ndprf  18127  uncfcl  18156  uncf1  18157  uncf2  18158  yonedalem1  18193  yonedalem21  18194  yonedalem22  18199  funcrngcsetcALT  20572  rescofuf  49280  cofu1a  49281  cofu2a  49282  cofucla  49283  cofuoppf  49337  uptrlem2  49398  uptra  49402  uptr2a  49409  cofuswapfcl  49480  prcofdiag1  49580  prcofdiag  49581  oppfdiag1  49601  oppfdiag  49603  cofuterm  49732