Theorem funcres 17991

Description: A functor restricted to a subcategory is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
funcres.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
funcres.h	⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶))

Assertion

Ref	Expression
funcres	⊢ (𝜑 → (𝐹 ↾f 𝐻) ∈ ((𝐶 ↾cat 𝐻) Func 𝐷))

Proof of Theorem funcres

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

Step	Hyp	Ref	Expression
1		funcres.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
2		funcres.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶))
3	1, 2	resfval 17987	. . 3 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
4	3	fveq2d 6886	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) = (2nd ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩))
5		fvex 6895	. . . . . . 7 ⊢ (1st ‘𝐹) ∈ V
6	5	resex 6026	. . . . . 6 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V
7		dmexg 7902	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
8		mptexg 7224	. . . . . . 7 ⊢ (dom 𝐻 ∈ V → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V)
9	2, 7, 8	3syl 19	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V)
10		op2ndg 8003	. . . . . 6 ⊢ ((((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V ∧ (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V) → (2nd ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))))
11	6, 9, 10	sylancr 599	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))))
12	4, 11	eqtrd 2797	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))))
13	12	opeq2d 4843	. . 3 ⊢ (𝜑 → ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹 ↾f 𝐻))⟩ = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
14	3, 13	eqtr4d 2800	. 2 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹 ↾f 𝐻))⟩)
15		eqid 2762	. . . 4 ⊢ (Base‘(𝐶 ↾cat 𝐻)) = (Base‘(𝐶 ↾cat 𝐻))
16		eqid 2762	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
17		eqid 2762	. . . 4 ⊢ (Hom ‘(𝐶 ↾cat 𝐻)) = (Hom ‘(𝐶 ↾cat 𝐻))
18		eqid 2762	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
19		eqid 2762	. . . 4 ⊢ (Id‘(𝐶 ↾cat 𝐻)) = (Id‘(𝐶 ↾cat 𝐻))
20		eqid 2762	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
21		eqid 2762	. . . 4 ⊢ (comp‘(𝐶 ↾cat 𝐻)) = (comp‘(𝐶 ↾cat 𝐻))
22		eqid 2762	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
23		eqid 2762	. . . . 5 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻)
24	23, 2	subccat 17943	. . . 4 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) ∈ Cat)
25		funcrcl 17958	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
26	1, 25	syl 18	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
27	26	simprd 501	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
28		eqid 2762	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
29		relfunc 17957	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
30		1st2ndbr 8043	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
31	29, 1, 30	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
32	28, 16, 31	funcf1 17961	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
33		eqidd 2763	. . . . . . . 8 ⊢ (𝜑 → dom dom 𝐻 = dom dom 𝐻)
34	2, 33	subcfn 17936	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
35	2, 34, 28	subcss1 17937	. . . . . 6 ⊢ (𝜑 → dom dom 𝐻 ⊆ (Base‘𝐶))
36	32, 35	fssresd 6746	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻):dom dom 𝐻⟶(Base‘𝐷))
37	26	simpld 500	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
38	23, 28, 37, 34, 35	rescbas 17924	. . . . . 6 ⊢ (𝜑 → dom dom 𝐻 = (Base‘(𝐶 ↾cat 𝐻)))
39	38	feq2d 6690	. . . . 5 ⊢ (𝜑 → (((1st ‘𝐹) ↾ dom dom 𝐻):dom dom 𝐻⟶(Base‘𝐷) ↔ ((1st ‘𝐹) ↾ dom dom 𝐻):(Base‘(𝐶 ↾cat 𝐻))⟶(Base‘𝐷)))
40	36, 39	mpbid 235	. . . 4 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻):(Base‘(𝐶 ↾cat 𝐻))⟶(Base‘𝐷))
41		fvex 6895	. . . . . . 7 ⊢ ((2nd ‘𝐹)‘𝑧) ∈ V
42	41	resex 6026	. . . . . 6 ⊢ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)) ∈ V
43		eqid 2762	. . . . . 6 ⊢ (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))
44	42, 43	fnmpti 6679	. . . . 5 ⊢ (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) Fn dom 𝐻
45	12	eqcomd 2768	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) = (2nd ‘(𝐹 ↾f 𝐻)))
46		fndm 6639	. . . . . . . 8 ⊢ (𝐻 Fn (dom dom 𝐻 × dom dom 𝐻) → dom 𝐻 = (dom dom 𝐻 × dom dom 𝐻))
47	34, 46	syl 18	. . . . . . 7 ⊢ (𝜑 → dom 𝐻 = (dom dom 𝐻 × dom dom 𝐻))
48	38	sqxpeqd 5691	. . . . . . 7 ⊢ (𝜑 → (dom dom 𝐻 × dom dom 𝐻) = ((Base‘(𝐶 ↾cat 𝐻)) × (Base‘(𝐶 ↾cat 𝐻))))
49	47, 48	eqtrd 2797	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = ((Base‘(𝐶 ↾cat 𝐻)) × (Base‘(𝐶 ↾cat 𝐻))))
50	45, 49	fneq12d 6631	. . . . 5 ⊢ (𝜑 → ((𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) Fn dom 𝐻 ↔ (2nd ‘(𝐹 ↾f 𝐻)) Fn ((Base‘(𝐶 ↾cat 𝐻)) × (Base‘(𝐶 ↾cat 𝐻)))))
51	44, 50	mpbii 236	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) Fn ((Base‘(𝐶 ↾cat 𝐻)) × (Base‘(𝐶 ↾cat 𝐻))))
52		eqid 2762	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
53	31	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
54	35	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → dom dom 𝐻 ⊆ (Base‘𝐶))
55		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)))
56	38	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → dom dom 𝐻 = (Base‘(𝐶 ↾cat 𝐻)))
57	55, 56	eleqtrrd 2865	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑥 ∈ dom dom 𝐻)
58	54, 57	sseldd 3935	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑥 ∈ (Base‘𝐶))
59		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))
60	59, 56	eleqtrrd 2865	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑦 ∈ dom dom 𝐻)
61	54, 60	sseldd 3935	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑦 ∈ (Base‘𝐶))
62	28, 52, 18, 53, 58, 61	funcf2 17963	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
63	2	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝐻 ∈ (Subcat‘𝐶))
64	34	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
65	63, 64, 52, 57, 60	subcss2 17938	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥𝐻𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦))
66	62, 65	fssresd 6746	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → ((𝑥(2nd ‘𝐹)𝑦) ↾ (𝑥𝐻𝑦)):(𝑥𝐻𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
67	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝐹 ∈ (𝐶 Func 𝐷))
68	67, 63, 64, 57, 60	resf2nd 17990	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦) = ((𝑥(2nd ‘𝐹)𝑦) ↾ (𝑥𝐻𝑦)))
69	68	feq1d 6688	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦):(𝑥𝐻𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)) ↔ ((𝑥(2nd ‘𝐹)𝑦) ↾ (𝑥𝐻𝑦)):(𝑥𝐻𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))))
70	66, 69	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦):(𝑥𝐻𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
71	23, 28, 37, 34, 35	reschom 17925	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = (Hom ‘(𝐶 ↾cat 𝐻)))
72	71	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝐻 = (Hom ‘(𝐶 ↾cat 𝐻)))
73	72	oveqd 7434	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥𝐻𝑦) = (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦))
74	57	fvresd 6902	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st ‘𝐹)‘𝑥))
75	60	fvresd 6902	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦) = ((1st ‘𝐹)‘𝑦))
76	74, 75	oveq12d 7435	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → ((((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥)(Hom ‘𝐷)(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦)) = (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
77	76	eqcomd 2768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)) = ((((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥)(Hom ‘𝐷)(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦)))
78	73, 77	feq23d 6701	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦):(𝑥𝐻𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)) ↔ (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦):(𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦)⟶((((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥)(Hom ‘𝐷)(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦))))
79	70, 78	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦):(𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦)⟶((((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥)(Hom ‘𝐷)(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦)))
80	1	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝐹 ∈ (𝐶 Func 𝐷))
81	2	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝐻 ∈ (Subcat‘𝐶))
82	34	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
83	38	eleq2d 2848	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ dom dom 𝐻 ↔ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))))
84	83	biimpar 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝑥 ∈ dom dom 𝐻)
85	80, 81, 82, 84, 84	resf2nd 17990	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑥) = ((𝑥(2nd ‘𝐹)𝑥) ↾ (𝑥𝐻𝑥)))
86		eqid 2762	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
87	23, 81, 82, 86, 84	subcid 17942	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((Id‘𝐶)‘𝑥) = ((Id‘(𝐶 ↾cat 𝐻))‘𝑥))
88	87	eqcomd 2768	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((Id‘(𝐶 ↾cat 𝐻))‘𝑥) = ((Id‘𝐶)‘𝑥))
89	85, 88	fveq12d 6889	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑥)‘((Id‘(𝐶 ↾cat 𝐻))‘𝑥)) = (((𝑥(2nd ‘𝐹)𝑥) ↾ (𝑥𝐻𝑥))‘((Id‘𝐶)‘𝑥)))
90	31	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
91	38, 35	eqsstrrd 3969	. . . . . . . 8 ⊢ (𝜑 → (Base‘(𝐶 ↾cat 𝐻)) ⊆ (Base‘𝐶))
92	91	sselda 3934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝑥 ∈ (Base‘𝐶))
93	28, 86, 20, 90, 92	funcid 17965	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((𝑥(2nd ‘𝐹)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘((1st ‘𝐹)‘𝑥)))
94	81, 82, 84, 86	subcidcl 17939	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))
95	94	fvresd 6902	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (((𝑥(2nd ‘𝐹)𝑥) ↾ (𝑥𝐻𝑥))‘((Id‘𝐶)‘𝑥)) = ((𝑥(2nd ‘𝐹)𝑥)‘((Id‘𝐶)‘𝑥)))
96	84	fvresd 6902	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st ‘𝐹)‘𝑥))
97	96	fveq2d 6886	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((Id‘𝐷)‘(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥)) = ((Id‘𝐷)‘((1st ‘𝐹)‘𝑥)))
98	93, 95, 97	3eqtr4d 2807	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (((𝑥(2nd ‘𝐹)𝑥) ↾ (𝑥𝐻𝑥))‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥)))
99	89, 98	eqtrd 2797	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑥)‘((Id‘(𝐶 ↾cat 𝐻))‘𝑥)) = ((Id‘𝐷)‘(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥)))
100	2	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝐻 ∈ (Subcat‘𝐶))
101	34	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
102		simp21 1225	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)))
103	38	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → dom dom 𝐻 = (Base‘(𝐶 ↾cat 𝐻)))
104	102, 103	eleqtrrd 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑥 ∈ dom dom 𝐻)
105		eqid 2762	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
106		simp22 1226	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))
107	106, 103	eleqtrrd 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑦 ∈ dom dom 𝐻)
108		simp23 1227	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻)))
109	108, 103	eleqtrrd 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑧 ∈ dom dom 𝐻)
110		simp3l 1220	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦))
111	71	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝐻 = (Hom ‘(𝐶 ↾cat 𝐻)))
112	111	oveqd 7434	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑥𝐻𝑦) = (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦))
113	110, 112	eleqtrrd 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑓 ∈ (𝑥𝐻𝑦))
114		simp3r 1221	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))
115	111	oveqd 7434	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑦𝐻𝑧) = (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))
116	114, 115	eleqtrrd 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑔 ∈ (𝑦𝐻𝑧))
117	100, 101, 104, 105, 107, 109, 113, 116	subccocl 17940	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))
118	117	fvresd 6902	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((𝑥(2nd ‘𝐹)𝑧) ↾ (𝑥𝐻𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = ((𝑥(2nd ‘𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))
119	31	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
120	35	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → dom dom 𝐻 ⊆ (Base‘𝐶))
121	120, 104	sseldd 3935	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑥 ∈ (Base‘𝐶))
122	120, 107	sseldd 3935	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑦 ∈ (Base‘𝐶))
123	120, 109	sseldd 3935	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑧 ∈ (Base‘𝐶))
124	100, 101, 52, 104, 107	subcss2 17938	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑥𝐻𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦))
125	124, 113	sseldd 3935	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
126	100, 101, 52, 107, 109	subcss2 17938	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑦𝐻𝑧) ⊆ (𝑦(Hom ‘𝐶)𝑧))
127	126, 116	sseldd 3935	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
128	28, 52, 105, 22, 119, 121, 122, 123, 125, 127	funcco 17966	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))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)))
129	118, 128	eqtrd 2797	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((𝑥(2nd ‘𝐹)𝑧) ↾ (𝑥𝐻𝑧))‘(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)))
130	1	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝐹 ∈ (𝐶 Func 𝐷))
131	130, 100, 101, 104, 109	resf2nd 17990	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑧) = ((𝑥(2nd ‘𝐹)𝑧) ↾ (𝑥𝐻𝑧)))
132	23, 28, 37, 34, 35, 105	rescco 17927	. . . . . . . . . 10 ⊢ (𝜑 → (comp‘𝐶) = (comp‘(𝐶 ↾cat 𝐻)))
133	132	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (comp‘𝐶) = (comp‘(𝐶 ↾cat 𝐻)))
134	133	eqcomd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (comp‘(𝐶 ↾cat 𝐻)) = (comp‘𝐶))
135	134	oveqd 7434	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾cat 𝐻))𝑧) = (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧))
136	135	oveqd 7434	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾cat 𝐻))𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))
137	131, 136	fveq12d 6889	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾cat 𝐻))𝑧)𝑓)) = (((𝑥(2nd ‘𝐹)𝑧) ↾ (𝑥𝐻𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))
138	104	fvresd 6902	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st ‘𝐹)‘𝑥))
139	107	fvresd 6902	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦) = ((1st ‘𝐹)‘𝑦))
140	138, 139	opeq12d 4844	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → ⟨(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥), (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦)⟩ = ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩)
141	109	fvresd 6902	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑧) = ((1st ‘𝐹)‘𝑧))
142	140, 141	oveq12d 7435	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (⟨(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥), (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦)⟩(comp‘𝐷)(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑧)) = (⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐹)‘𝑧)))
143	130, 100, 101, 107, 109	resf2nd 17990	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑦(2nd ‘(𝐹 ↾f 𝐻))𝑧) = ((𝑦(2nd ‘𝐹)𝑧) ↾ (𝑦𝐻𝑧)))
144	143	fveq1d 6884	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → ((𝑦(2nd ‘(𝐹 ↾f 𝐻))𝑧)‘𝑔) = (((𝑦(2nd ‘𝐹)𝑧) ↾ (𝑦𝐻𝑧))‘𝑔))
145	116	fvresd 6902	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((𝑦(2nd ‘𝐹)𝑧) ↾ (𝑦𝐻𝑧))‘𝑔) = ((𝑦(2nd ‘𝐹)𝑧)‘𝑔))
146	144, 145	eqtrd 2797	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → ((𝑦(2nd ‘(𝐹 ↾f 𝐻))𝑧)‘𝑔) = ((𝑦(2nd ‘𝐹)𝑧)‘𝑔))
147	130, 100, 101, 104, 107	resf2nd 17990	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦) = ((𝑥(2nd ‘𝐹)𝑦) ↾ (𝑥𝐻𝑦)))
148	147	fveq1d 6884	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦)‘𝑓) = (((𝑥(2nd ‘𝐹)𝑦) ↾ (𝑥𝐻𝑦))‘𝑓))
149	113	fvresd 6902	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((𝑥(2nd ‘𝐹)𝑦) ↾ (𝑥𝐻𝑦))‘𝑓) = ((𝑥(2nd ‘𝐹)𝑦)‘𝑓))
150	148, 149	eqtrd 2797	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦)‘𝑓) = ((𝑥(2nd ‘𝐹)𝑦)‘𝑓))
151	142, 146, 150	oveq123d 7438	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((𝑦(2nd ‘(𝐹 ↾f 𝐻))𝑧)‘𝑔)(⟨(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥), (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦)⟩(comp‘𝐷)(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑧))((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦)‘𝑓)) = (((𝑦(2nd ‘𝐹)𝑧)‘𝑔)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐹)‘𝑧))((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
152	129, 137, 151	3eqtr4d 2807	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾cat 𝐻))𝑧)𝑓)) = (((𝑦(2nd ‘(𝐹 ↾f 𝐻))𝑧)‘𝑔)(⟨(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥), (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦)⟩(comp‘𝐷)(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑧))((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦)‘𝑓)))
153	15, 16, 17, 18, 19, 20, 21, 22, 24, 27, 40, 51, 79, 99, 152	isfuncd 17960	. . 3 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻)((𝐶 ↾cat 𝐻) Func 𝐷)(2nd ‘(𝐹 ↾f 𝐻)))
154		df-br 5108	. . 3 ⊢ (((1st ‘𝐹) ↾ dom dom 𝐻)((𝐶 ↾cat 𝐻) Func 𝐷)(2nd ‘(𝐹 ↾f 𝐻)) ↔ ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹 ↾f 𝐻))⟩ ∈ ((𝐶 ↾cat 𝐻) Func 𝐷))
155	153, 154	sylib 221	. 2 ⊢ (𝜑 → ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹 ↾f 𝐻))⟩ ∈ ((𝐶 ↾cat 𝐻) Func 𝐷))
156	14, 155	eqeltrd 2862	1 ⊢ (𝜑 → (𝐹 ↾f 𝐻) ∈ ((𝐶 ↾cat 𝐻) Func 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902  ⟨cop 4593   class class class wbr 5107   ↦ cmpt 5190   × cxp 5657  dom cdm 5659   ↾ cres 5661  Rel wrel 5664   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417  1^st c1st 7988  2^nd c2nd 7989  Basecbs 17307  Hom chom 17359  compcco 17360  Catccat 17758  Idccid 17759   ↾_cat cresc 17903  Subcatcsubc 17904   Func cfunc 17949   ↾_f cresf 17952

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-9 12338  df-n0 12533  df-z 12620  df-dec 12741  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-hom 17372  df-cco 17373  df-cat 17762  df-cid 17763  df-homf 17764  df-ssc 17905  df-resc 17906  df-subc 17907  df-func 17953  df-resf 17956

This theorem is used by:  funcrngcsetc  20808  funcringcsetc  20842