Theorem resssetc 18059

Description: The restriction of the category of sets to a subset is the category of sets in the subset. Thus, the SetCat‘𝑈 categories for different 𝑈 are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
resssetc.c	⊢ 𝐶 = (SetCat‘𝑈)
resssetc.d	⊢ 𝐷 = (SetCat‘𝑉)
resssetc.1	⊢ (𝜑 → 𝑈 ∈ 𝑊)
resssetc.2	⊢ (𝜑 → 𝑉 ⊆ 𝑈)

Assertion

Ref	Expression
resssetc	⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ∧ (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷)))

Proof of Theorem resssetc

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

Step	Hyp	Ref	Expression
1		resssetc.d	. . . . . 6 ⊢ 𝐷 = (SetCat‘𝑉)
2		resssetc.1	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
3		resssetc.2	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
4	2, 3	ssexd 5265	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ V)
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑉 ∈ V)
6		eqid 2736	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
8		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ 𝑉)
9	1, 5, 6, 7, 8	setchom 18047	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑦 ↑m 𝑥))
10		resssetc.c	. . . . . 6 ⊢ 𝐶 = (SetCat‘𝑈)
11	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑈 ∈ 𝑊)
12		eqid 2736	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
13	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑉 ⊆ 𝑈)
14	13, 7	sseldd 3922	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝑈)
15	13, 8	sseldd 3922	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ 𝑈)
16	10, 11, 12, 14, 15	setchom 18047	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑦 ↑m 𝑥))
17		eqid 2736	. . . . . . . 8 ⊢ (𝐶 ↾s 𝑉) = (𝐶 ↾s 𝑉)
18	17, 12	resshom 17381	. . . . . . 7 ⊢ (𝑉 ∈ V → (Hom ‘𝐶) = (Hom ‘(𝐶 ↾s 𝑉)))
19	4, 18	syl 17	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘(𝐶 ↾s 𝑉)))
20	19	oveqdr 7395	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦))
21	9, 16, 20	3eqtr2rd 2778	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
22	21	ralrimivva 3180	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
23		eqid 2736	. . . 4 ⊢ (Hom ‘(𝐶 ↾s 𝑉)) = (Hom ‘(𝐶 ↾s 𝑉))
24	10, 2	setcbas 18045	. . . . . 6 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
25	3, 24	sseqtrd 3958	. . . . 5 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐶))
26		eqid 2736	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
27	17, 26	ressbas2 17208	. . . . 5 ⊢ (𝑉 ⊆ (Base‘𝐶) → 𝑉 = (Base‘(𝐶 ↾s 𝑉)))
28	25, 27	syl 17	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘(𝐶 ↾s 𝑉)))
29	1, 4	setcbas 18045	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝐷))
30	23, 6, 28, 29	homfeq 17660	. . 3 ⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
31	22, 30	mpbird 257	. 2 ⊢ (𝜑 → (Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷))
32	4	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉 ∈ V)
33		eqid 2736	. . . . . . . 8 ⊢ (comp‘𝐷) = (comp‘𝐷)
34		simplr1 1217	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥 ∈ 𝑉)
35		simplr2 1218	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦 ∈ 𝑉)
36		simplr3 1219	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧 ∈ 𝑉)
37		simprl 771	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
38	1, 32, 6, 34, 35	elsetchom 18048	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ↔ 𝑓:𝑥⟶𝑦))
39	37, 38	mpbid 232	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓:𝑥⟶𝑦)
40		simprr 773	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
41	1, 32, 6, 35, 36	elsetchom 18048	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↔ 𝑔:𝑦⟶𝑧))
42	40, 41	mpbid 232	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔:𝑦⟶𝑧)
43	1, 32, 33, 34, 35, 36, 39, 42	setcco 18050	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔 ∘ 𝑓))
44	2	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑈 ∈ 𝑊)
45		eqid 2736	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
46	3	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉 ⊆ 𝑈)
47	46, 34	sseldd 3922	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥 ∈ 𝑈)
48	46, 35	sseldd 3922	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦 ∈ 𝑈)
49	46, 36	sseldd 3922	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧 ∈ 𝑈)
50	10, 44, 45, 47, 48, 49, 39, 42	setcco 18050	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔 ∘ 𝑓))
51	17, 45	ressco 17382	. . . . . . . . . . 11 ⊢ (𝑉 ∈ V → (comp‘𝐶) = (comp‘(𝐶 ↾s 𝑉)))
52	4, 51	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (comp‘𝐶) = (comp‘(𝐶 ↾s 𝑉)))
53	52	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (comp‘𝐶) = (comp‘(𝐶 ↾s 𝑉)))
54	53	oveqd 7384	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧) = (⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾s 𝑉))𝑧))
55	54	oveqd 7384	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾s 𝑉))𝑧)𝑓))
56	43, 50, 55	3eqtr2d 2777	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾s 𝑉))𝑧)𝑓))
57	56	ralrimivva 3180	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾s 𝑉))𝑧)𝑓))
58	57	ralrimivvva 3183	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ∀𝑧 ∈ 𝑉 ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾s 𝑉))𝑧)𝑓))
59		eqid 2736	. . . . 5 ⊢ (comp‘(𝐶 ↾s 𝑉)) = (comp‘(𝐶 ↾s 𝑉))
60	31	eqcomd 2742	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾s 𝑉)))
61	33, 59, 6, 29, 28, 60	comfeq 17672	. . . 4 ⊢ (𝜑 → ((compf‘𝐷) = (compf‘(𝐶 ↾s 𝑉)) ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ∀𝑧 ∈ 𝑉 ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾s 𝑉))𝑧)𝑓)))
62	58, 61	mpbird 257	. . 3 ⊢ (𝜑 → (compf‘𝐷) = (compf‘(𝐶 ↾s 𝑉)))
63	62	eqcomd 2742	. 2 ⊢ (𝜑 → (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷))
64	31, 63	jca 511	1 ⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ∧ (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ⊆ wss 3889  ⟨cop 4573   ∘ ccom 5635  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773  Basecbs 17179   ↾_s cress 17200  Hom chom 17231  compcco 17232  Hom_f chomf 17632  comp_fccomf 17633  SetCatcsetc 18042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-hom 17244  df-cco 17245  df-homf 17636  df-comf 17637  df-setc 18043

This theorem is referenced by:  funcsetcres2  18060