Theorem cofucl 17857

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 2730	. . . 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 17851	. . 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 17852	. . . 4 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = ((1st ‘𝐺) ∘ (1st ‘𝐹)))
6	4	fveq2d 6865	. . . . 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 6874	. . . . . . 7 ⊢ (1st ‘𝐺) ∈ V
8		fvex 6874	. . . . . . 7 ⊢ (1st ‘𝐹) ∈ V
9	7, 8	coex 7909	. . . . . 6 ⊢ ((1st ‘𝐺) ∘ (1st ‘𝐹)) ∈ V
10		fvex 6874	. . . . . . 7 ⊢ (Base‘𝐶) ∈ V
11	10, 10	mpoex 8061	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) ∈ V
12	9, 11	op2nd 7980	. . . . 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 2781	. . . 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 4848	. . 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 2768	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩)
16		eqid 2730	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
17		eqid 2730	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
18		relfunc 17831	. . . . . . . 8 ⊢ Rel (𝐷 Func 𝐸)
19		1st2ndbr 8024	. . . . . . . 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 17835	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐺):(Base‘𝐷)⟶(Base‘𝐸))
22		relfunc 17831	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
23		1st2ndbr 8024	. . . . . . . 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 17835	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
26		fco 6715	. . . . . 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 6673	. . . . 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 2730	. . . . . . 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 7423	. . . . . . . 8 ⊢ (((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∈ V
32		ovex 7423	. . . . . . . 8 ⊢ (𝑥(2nd ‘𝐹)𝑦) ∈ V
33	31, 32	coex 7909	. . . . . . 7 ⊢ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)) ∈ V
34	30, 33	fnmpoi 8052	. . . . . 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 6614	. . . . . 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 2730	. . . . . . . . . . 11 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
38		eqid 2730	. . . . . . . . . . 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 7060	. . . . . . . . . . 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 7060	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
45	16, 37, 38, 39, 42, 44	funcf2 17837	. . . . . . . . . 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 2730	. . . . . . . . . . 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 17837	. . . . . . . . . 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 6715	. . . . . . . . . 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 7423	. . . . . . . . . 10 ⊢ (((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))) ∈ V
52		ovex 7423	. . . . . . . . . 10 ⊢ (𝑥(Hom ‘𝐶)𝑦) ∈ V
53	51, 52	elmap 8847	. . . . . . . . 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 17854	. . . . . . . 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 17853	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑥) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)))
59	1, 55, 56, 43	cofu1 17853	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑦) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦)))
60	58, 59	oveq12d 7408	. . . . . . . . 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 7405	. . . . . . . 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 2844	. . . . . . 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 3181	. . . . . 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 6861	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd ‘(𝐺 ∘func 𝐹))‘𝑧) = ((2nd ‘(𝐺 ∘func 𝐹))‘⟨𝑥, 𝑦⟩))
65		df-ov 7393	. . . . . . . . 9 ⊢ (𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦) = ((2nd ‘(𝐺 ∘func 𝐹))‘⟨𝑥, 𝑦⟩)
66	64, 65	eqtr4di 2783	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd ‘(𝐺 ∘func 𝐹))‘𝑧) = (𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦))
67		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
68		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
69	67, 68	op1std 7981	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
70	69	fveq2d 6865	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st ‘(𝐺 ∘func 𝐹))‘(1st ‘𝑧)) = ((1st ‘(𝐺 ∘func 𝐹))‘𝑥))
71	67, 68	op2ndd 7982	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
72	71	fveq2d 6865	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st ‘(𝐺 ∘func 𝐹))‘(2nd ‘𝑧)) = ((1st ‘(𝐺 ∘func 𝐹))‘𝑦))
73	70, 72	oveq12d 7408	. . . . . . . . 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 6861	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((Hom ‘𝐶)‘𝑧) = ((Hom ‘𝐶)‘⟨𝑥, 𝑦⟩))
75		df-ov 7393	. . . . . . . . . 10 ⊢ (𝑥(Hom ‘𝐶)𝑦) = ((Hom ‘𝐶)‘⟨𝑥, 𝑦⟩)
76	74, 75	eqtr4di 2783	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((Hom ‘𝐶)‘𝑧) = (𝑥(Hom ‘𝐶)𝑦))
77	73, 76	oveq12d 7408	. . . . . . . 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 2823	. . . . . . 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 5808	. . . . . 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 6874	. . . . . 6 ⊢ (2nd ‘(𝐺 ∘func 𝐹)) ∈ V
82	81	elixp 8880	. . . . 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 2730	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
85		eqid 2730	. . . . . . . . . 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 17839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd ‘𝐹)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘((1st ‘𝐹)‘𝑥)))
89	88	fveq2d 6865	. . . . . . . 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 2730	. . . . . . . . 9 ⊢ (Id‘𝐸) = (Id‘𝐸)
91	20	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
92	25	ffvelcdmda 7059	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
93	16, 85, 90, 91, 92	funcid 17839	. . . . . . . 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 2765	. . . . . . 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 17832	. . . . . . . . . . . 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 17650	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
102	1, 95, 96, 87, 87, 46, 101	cofu2 17855	. . . . . . 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 17853	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑥) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)))
104	103	fveq2d 6865	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐸)‘((1st ‘(𝐺 ∘func 𝐹))‘𝑥)) = ((Id‘𝐸)‘((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))))
105	94, 102, 104	3eqtr4d 2775	. . . . . 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 17837	. . . . . . . . . . . 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 2730	. . . . . . . . . . . . 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 17653	. . . . . . . . . . . 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 6963	. . . . . . . . . . . 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 2730	. . . . . . . . . . . . 13 ⊢ (comp‘𝐷) = (comp‘𝐷)
119	1, 46, 110, 118, 106, 107, 112, 108, 113, 114	funcco 17840	. . . . . . . . . . . 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 6865	. . . . . . . . . . 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 2730	. . . . . . . . . . . 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 7060	. . . . . . . . . . . 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 7060	. . . . . . . . . . . 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 17837	. . . . . . . . . . . . 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 7060	. . . . . . . . . . . 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 17837	. . . . . . . . . . . . 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 7060	. . . . . . . . . . . 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 17840	. . . . . . . . . . 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 2769	. . . . . . . . . 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 17854	. . . . . . . . . . 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 6863	. . . . . . . . . 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 17853	. . . . . . . . . . . . 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 4848	. . . . . . . . . . . 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 17853	. . . . . . . . . . . 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 7408	. . . . . . . . . . 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 17855	. . . . . . . . . . 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 17855	. . . . . . . . . . 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 7411	. . . . . . . . . 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 2775	. . . . . . . . 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 3181	. . . . . . 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 3181	. . . . . 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 17832	. . . . . . 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 17833	. . . 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 5111	. . 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 2829	1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Func 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ⟨cop 4598   class class class wbr 5110   × cxp 5639   ∘ ccom 5645  Rel wrel 5646   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  1^st c1st 7969  2^nd c2nd 7970   ↑_m cmap 8802  Xcixp 8873  Basecbs 17186  Hom chom 17238  compcco 17239  Catccat 17632  Idccid 17633   Func cfunc 17823   ∘_func ccofu 17825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-ixp 8874  df-cat 17636  df-cid 17637  df-func 17827  df-cofu 17829

This theorem is referenced by:  cofuass  17858  cofull  17905  cofth  17906  catccatid  18075  1st2ndprf  18174  uncfcl  18203  uncf1  18204  uncf2  18205  yonedalem1  18240  yonedalem21  18241  yonedalem22  18246  funcrngcsetcALT  20557  rescofuf  49086  cofu1a  49087  cofu2a  49088  cofucla  49089  cofuoppf  49143  uptrlem2  49204  uptra  49208  uptr2a  49215  cofuswapfcl  49286  prcofdiag1  49386  prcofdiag  49387  oppfdiag1  49407  oppfdiag  49409  cofuterm  49538