Theorem funcres 18076

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 18072	. . 3 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
4	3	fveq2d 6889	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) = (2nd ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩))
5		fvex 6898	. . . . . . 7 ⊢ (1st ‘𝐹) ∈ V
6	5	resex 6017	. . . . . 6 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V
7		dmexg 7913	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
8		mptexg 7227	. . . . . . 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 8014	. . . . . 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 2795	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))))
13	12	opeq2d 4846	. . 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 2798	. 2 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹 ↾f 𝐻))⟩)
15		eqid 2760	. . . 4 ⊢ (Base‘(𝐶 ↾cat 𝐻)) = (Base‘(𝐶 ↾cat 𝐻))
16		eqid 2760	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
17		eqid 2760	. . . 4 ⊢ (Hom ‘(𝐶 ↾cat 𝐻)) = (Hom ‘(𝐶 ↾cat 𝐻))
18		eqid 2760	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
19		eqid 2760	. . . 4 ⊢ (Id‘(𝐶 ↾cat 𝐻)) = (Id‘(𝐶 ↾cat 𝐻))
20		eqid 2760	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
21		eqid 2760	. . . 4 ⊢ (comp‘(𝐶 ↾cat 𝐻)) = (comp‘(𝐶 ↾cat 𝐻))
22		eqid 2760	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
23		eqid 2760	. . . . 5 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻)
24	23, 2	subccat 18028	. . . 4 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) ∈ Cat)
25		funcrcl 18043	. . . . . 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 2760	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
29		relfunc 18042	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
30		1st2ndbr 8053	. . . . . . . 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 18046	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
33		eqidd 2761	. . . . . . . 8 ⊢ (𝜑 → dom dom 𝐻 = dom dom 𝐻)
34	2, 33	subcfn 18021	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
35	2, 34, 28	subcss1 18022	. . . . . 6 ⊢ (𝜑 → dom dom 𝐻 ⊆ (Base‘𝐶))
36	32, 35	fssresd 6749	. . . . 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 18009	. . . . . 6 ⊢ (𝜑 → dom dom 𝐻 = (Base‘(𝐶 ↾cat 𝐻)))
39	38	feq2d 6693	. . . . 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 6898	. . . . . . 7 ⊢ ((2nd ‘𝐹)‘𝑧) ∈ V
42	41	resex 6017	. . . . . 6 ⊢ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)) ∈ V
43		eqid 2760	. . . . . 6 ⊢ (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))
44	42, 43	fnmpti 6682	. . . . 5 ⊢ (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) Fn dom 𝐻
45	12	eqcomd 2766	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) = (2nd ‘(𝐹 ↾f 𝐻)))
46		fndm 6642	. . . . . . . 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 5682	. . . . . . 7 ⊢ (𝜑 → (dom dom 𝐻 × dom dom 𝐻) = ((Base‘(𝐶 ↾cat 𝐻)) × (Base‘(𝐶 ↾cat 𝐻))))
49	47, 48	eqtrd 2795	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = ((Base‘(𝐶 ↾cat 𝐻)) × (Base‘(𝐶 ↾cat 𝐻))))
50	45, 49	fneq12d 6634	. . . . 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 2760	. . . . . . . 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 2863	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑥 ∈ dom dom 𝐻)
58	54, 57	sseldd 3909	. . . . . . . 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 2863	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑦 ∈ dom dom 𝐻)
61	54, 60	sseldd 3909	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑦 ∈ (Base‘𝐶))
62	28, 52, 18, 53, 58, 61	funcf2 18048	. . . . . . 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 18023	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥𝐻𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦))
66	62, 65	fssresd 6749	. . . . . 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 18075	. . . . . . 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 6691	. . . . . 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 18010	. . . . . . . 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 7437	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥𝐻𝑦) = (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦))
74	57	fvresd 6905	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st ‘𝐹)‘𝑥))
75	60	fvresd 6905	. . . . . . . 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 7438	. . . . . . 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 2766	. . . . . 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 6704	. . . . 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 2846	. . . . . . . 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 18075	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑥) = ((𝑥(2nd ‘𝐹)𝑥) ↾ (𝑥𝐻𝑥)))
86		eqid 2760	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
87	23, 81, 82, 86, 84	subcid 18027	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((Id‘𝐶)‘𝑥) = ((Id‘(𝐶 ↾cat 𝐻))‘𝑥))
88	87	eqcomd 2766	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((Id‘(𝐶 ↾cat 𝐻))‘𝑥) = ((Id‘𝐶)‘𝑥))
89	85, 88	fveq12d 6892	. . . . 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 3943	. . . . . . . 8 ⊢ (𝜑 → (Base‘(𝐶 ↾cat 𝐻)) ⊆ (Base‘𝐶))
92	91	sselda 3908	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝑥 ∈ (Base‘𝐶))
93	28, 86, 20, 90, 92	funcid 18050	. . . . . 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 18024	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))
95	94	fvresd 6905	. . . . . 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 6905	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st ‘𝐹)‘𝑥))
97	96	fveq2d 6889	. . . . . 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 2805	. . . . 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 2795	. . . 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 2863	. . . . . . . 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 2760	. . . . . . . 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 2863	. . . . . . . 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 2863	. . . . . . . 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 7437	. . . . . . . . 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 2863	. . . . . . . 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 7437	. . . . . . . . 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 2863	. . . . . . . 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 18025	. . . . . . 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 6905	. . . . . 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 3909	. . . . . . 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 3909	. . . . . . 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 3909	. . . . . . 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 18023	. . . . . . . 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 3909	. . . . . . 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 18023	. . . . . . . 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 3909	. . . . . . 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 18051	. . . . . 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 2795	. . . . 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 18075	. . . . . 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 18012	. . . . . . . . . 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 2766	. . . . . . . 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 7437	. . . . . . 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 7437	. . . . . 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 6892	. . . . 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 6905	. . . . . . . 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 6905	. . . . . . . 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 4847	. . . . . . 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 6905	. . . . . . 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 7438	. . . . . 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 18075	. . . . . . . 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 6887	. . . . . . 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 6905	. . . . . . 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 2795	. . . . . 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 18075	. . . . . . . 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 6887	. . . . . . 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 6905	. . . . . . 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 2795	. . . . . 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 7441	. . . . 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 2805	. . . 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 18045	. . 3 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻)((𝐶 ↾cat 𝐻) Func 𝐷)(2nd ‘(𝐹 ↾f 𝐻)))
154		df-br 5103	. . 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 2860	1 ⊢ (𝜑 → (𝐹 ↾f 𝐻) ∈ ((𝐶 ↾cat 𝐻) Func 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895  ⟨cop 4834   class class class wbr 5102   ↦ cmpt 5185   × cxp 5648  dom cdm 5650   ↾ cres 5652  Rel wrel 5655   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17392  Hom chom 17444  compcco 17445  Catccat 17843  Idccid 17844   ↾_cat cresc 17988  Subcatcsubc 17989   Func cfunc 18034   ↾_f cresf 18037

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  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-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-er 8720  df-map 8852  df-pm 8853  df-ixp 8929  df-en 8977  df-dom 8978  df-sdom 8979  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-3 12409  df-4 12410  df-5 12411  df-6 12412  df-7 12413  df-8 12414  df-9 12415  df-n0 12610  df-z 12697  df-dec 12818  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-ress 17414  df-hom 17457  df-cco 17458  df-cat 17847  df-cid 17848  df-homf 17849  df-ssc 17990  df-resc 17991  df-subc 17992  df-func 18038  df-resf 18041

This theorem is used by:  funcrngcsetc  20897  funcringcsetc  20931