Theorem funcres 17024

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 17020	. . 3 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
4	3	fveq2d 6503	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) = (2nd ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩))
5		fvex 6512	. . . . . . 7 ⊢ (1st ‘𝐹) ∈ V
6	5	resex 5744	. . . . . 6 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V
7		dmexg 7428	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
8		mptexg 6810	. . . . . . 7 ⊢ (dom 𝐻 ∈ V → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V)
9	2, 7, 8	3syl 18	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V)
10		op2ndg 7514	. . . . . 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 578	. . . . 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 2814	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))))
13	12	opeq2d 4684	. . 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 2817	. 2 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹 ↾f 𝐻))⟩)
15		eqid 2778	. . . 4 ⊢ (Base‘(𝐶 ↾cat 𝐻)) = (Base‘(𝐶 ↾cat 𝐻))
16		eqid 2778	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
17		eqid 2778	. . . 4 ⊢ (Hom ‘(𝐶 ↾cat 𝐻)) = (Hom ‘(𝐶 ↾cat 𝐻))
18		eqid 2778	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
19		eqid 2778	. . . 4 ⊢ (Id‘(𝐶 ↾cat 𝐻)) = (Id‘(𝐶 ↾cat 𝐻))
20		eqid 2778	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
21		eqid 2778	. . . 4 ⊢ (comp‘(𝐶 ↾cat 𝐻)) = (comp‘(𝐶 ↾cat 𝐻))
22		eqid 2778	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
23		eqid 2778	. . . . 5 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻)
24	23, 2	subccat 16976	. . . 4 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) ∈ Cat)
25		funcrcl 16991	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
26	1, 25	syl 17	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
27	26	simprd 488	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
28		eqid 2778	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
29		relfunc 16990	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
30		1st2ndbr 7553	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
31	29, 1, 30	sylancr 578	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
32	28, 16, 31	funcf1 16994	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
33		eqidd 2779	. . . . . . . 8 ⊢ (𝜑 → dom dom 𝐻 = dom dom 𝐻)
34	2, 33	subcfn 16969	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
35	2, 34, 28	subcss1 16970	. . . . . 6 ⊢ (𝜑 → dom dom 𝐻 ⊆ (Base‘𝐶))
36	32, 35	fssresd 6374	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻):dom dom 𝐻⟶(Base‘𝐷))
37	26	simpld 487	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
38	23, 28, 37, 34, 35	rescbas 16957	. . . . . 6 ⊢ (𝜑 → dom dom 𝐻 = (Base‘(𝐶 ↾cat 𝐻)))
39	38	feq2d 6330	. . . . 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 224	. . . 4 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻):(Base‘(𝐶 ↾cat 𝐻))⟶(Base‘𝐷))
41		fvex 6512	. . . . . . 7 ⊢ ((2nd ‘𝐹)‘𝑧) ∈ V
42	41	resex 5744	. . . . . 6 ⊢ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)) ∈ V
43		eqid 2778	. . . . . 6 ⊢ (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))
44	42, 43	fnmpti 6321	. . . . 5 ⊢ (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) Fn dom 𝐻
45	12	eqcomd 2784	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) = (2nd ‘(𝐹 ↾f 𝐻)))
46		fndm 6288	. . . . . . . 8 ⊢ (𝐻 Fn (dom dom 𝐻 × dom dom 𝐻) → dom 𝐻 = (dom dom 𝐻 × dom dom 𝐻))
47	34, 46	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝐻 = (dom dom 𝐻 × dom dom 𝐻))
48	38	sqxpeqd 5439	. . . . . . 7 ⊢ (𝜑 → (dom dom 𝐻 × dom dom 𝐻) = ((Base‘(𝐶 ↾cat 𝐻)) × (Base‘(𝐶 ↾cat 𝐻))))
49	47, 48	eqtrd 2814	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = ((Base‘(𝐶 ↾cat 𝐻)) × (Base‘(𝐶 ↾cat 𝐻))))
50	45, 49	fneq12d 6281	. . . . 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 225	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) Fn ((Base‘(𝐶 ↾cat 𝐻)) × (Base‘(𝐶 ↾cat 𝐻))))
52		eqid 2778	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
53	31	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
54	35	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → dom dom 𝐻 ⊆ (Base‘𝐶))
55		simprl 758	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)))
56	38	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → dom dom 𝐻 = (Base‘(𝐶 ↾cat 𝐻)))
57	55, 56	eleqtrrd 2869	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑥 ∈ dom dom 𝐻)
58	54, 57	sseldd 3859	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑥 ∈ (Base‘𝐶))
59		simprr 760	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))
60	59, 56	eleqtrrd 2869	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑦 ∈ dom dom 𝐻)
61	54, 60	sseldd 3859	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑦 ∈ (Base‘𝐶))
62	28, 52, 18, 53, 58, 61	funcf2 16996	. . . . . . 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 473	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝐻 ∈ (Subcat‘𝐶))
64	34	adantr 473	. . . . . . . 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 16971	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥𝐻𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦))
66	62, 65	fssresd 6374	. . . . . 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 473	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝐹 ∈ (𝐶 Func 𝐷))
68	67, 63, 64, 57, 60	resf2nd 17023	. . . . . . 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 6329	. . . . . 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 249	. . . . 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 16958	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = (Hom ‘(𝐶 ↾cat 𝐻)))
72	71	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝐻 = (Hom ‘(𝐶 ↾cat 𝐻)))
73	72	oveqd 6993	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥𝐻𝑦) = (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦))
74	57	fvresd 6519	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st ‘𝐹)‘𝑥))
75	60	fvresd 6519	. . . . . . . 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 6994	. . . . . . 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 2784	. . . . . 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 6339	. . . . 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 224	. . . 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 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝐹 ∈ (𝐶 Func 𝐷))
81	2	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝐻 ∈ (Subcat‘𝐶))
82	34	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
83	38	eleq2d 2851	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ dom dom 𝐻 ↔ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))))
84	83	biimpar 470	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝑥 ∈ dom dom 𝐻)
85	80, 81, 82, 84, 84	resf2nd 17023	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑥) = ((𝑥(2nd ‘𝐹)𝑥) ↾ (𝑥𝐻𝑥)))
86		eqid 2778	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
87	23, 81, 82, 86, 84	subcid 16975	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((Id‘𝐶)‘𝑥) = ((Id‘(𝐶 ↾cat 𝐻))‘𝑥))
88	87	eqcomd 2784	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((Id‘(𝐶 ↾cat 𝐻))‘𝑥) = ((Id‘𝐶)‘𝑥))
89	85, 88	fveq12d 6506	. . . . 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 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
91	38, 35	eqsstr3d 3896	. . . . . . . 8 ⊢ (𝜑 → (Base‘(𝐶 ↾cat 𝐻)) ⊆ (Base‘𝐶))
92	91	sselda 3858	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝑥 ∈ (Base‘𝐶))
93	28, 86, 20, 90, 92	funcid 16998	. . . . . 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 16972	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))
95	94	fvresd 6519	. . . . . 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 6519	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st ‘𝐹)‘𝑥))
97	96	fveq2d 6503	. . . . . 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 2824	. . . . 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 2814	. . . 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 1113	. . . . . . . 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 1113	. . . . . . . 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 1186	. . . . . . . . 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 1113	. . . . . . . . 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 2869	. . . . . . . 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 2778	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
106		simp22 1187	. . . . . . . . 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 2869	. . . . . . . 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 1188	. . . . . . . . 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 2869	. . . . . . . 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 1181	. . . . . . . . 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 1113	. . . . . . . . . 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 6993	. . . . . . . . 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 2869	. . . . . . . 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 1182	. . . . . . . . 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 6993	. . . . . . . . 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 2869	. . . . . . . 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 16973	. . . . . . 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 6519	. . . . . 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 1113	. . . . . . 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 1113	. . . . . . . 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 3859	. . . . . . 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 3859	. . . . . . 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 3859	. . . . . . 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 16971	. . . . . . . 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 3859	. . . . . . 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 16971	. . . . . . . 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 3859	. . . . . . 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 16999	. . . . . 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 2814	. . . . 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 1113	. . . . . . 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 17023	. . . . . 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 16960	. . . . . . . . . 10 ⊢ (𝜑 → (comp‘𝐶) = (comp‘(𝐶 ↾cat 𝐻)))
133	132	3ad2ant1 1113	. . . . . . . . 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 2784	. . . . . . . 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 6993	. . . . . . 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 6993	. . . . . 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 6506	. . . . 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 6519	. . . . . . . 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 6519	. . . . . . . 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 4685	. . . . . . 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 6519	. . . . . . 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 6994	. . . . . 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 17023	. . . . . . . 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 6501	. . . . . . 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 6519	. . . . . . 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 2814	. . . . . 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 17023	. . . . . . . 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 6501	. . . . . . 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 6519	. . . . . . 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 2814	. . . . . 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 6997	. . . . 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 2824	. . . 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 16993	. . 3 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻)((𝐶 ↾cat 𝐻) Func 𝐷)(2nd ‘(𝐹 ↾f 𝐻)))
154		df-br 4930	. . 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 210	. 2 ⊢ (𝜑 → ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹 ↾f 𝐻))⟩ ∈ ((𝐶 ↾cat 𝐻) Func 𝐷))
156	14, 155	eqeltrd 2866	1 ⊢ (𝜑 → (𝐹 ↾f 𝐻) ∈ ((𝐶 ↾cat 𝐻) Func 𝐷))

Syntax hints:   → wi 4   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050  Vcvv 3415   ⊆ wss 3829  ⟨cop 4447   class class class wbr 4929   ↦ cmpt 5008   × cxp 5405  dom cdm 5407   ↾ cres 5409  Rel wrel 5412   Fn wfn 6183  ⟶wf 6184  ‘cfv 6188  (class class class)co 6976  1^st c1st 7499  2^nd c2nd 7500  Basecbs 16339  Hom chom 16432  compcco 16433  Catccat 16793  Idccid 16794   ↾_cat cresc 16936  Subcatcsubc 16937   Func cfunc 16982   ↾_f cresf 16985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-er 8089  df-map 8208  df-pm 8209  df-ixp 8260  df-en 8307  df-dom 8308  df-sdom 8309  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-nn 11440  df-2 11503  df-3 11504  df-4 11505  df-5 11506  df-6 11507  df-7 11508  df-8 11509  df-9 11510  df-n0 11708  df-z 11794  df-dec 11912  df-ndx 16342  df-slot 16343  df-base 16345  df-sets 16346  df-ress 16347  df-hom 16445  df-cco 16446  df-cat 16797  df-cid 16798  df-homf 16799  df-ssc 16938  df-resc 16939  df-subc 16940  df-func 16986  df-resf 16989

This theorem is referenced by:  funcrngcsetc  43639  funcringcsetc  43676