Theorem resssetc 18090

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 5328	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ V)
5	4	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑉 ∈ V)
6		eqid 2728	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7		simprl 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
8		simprr 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ 𝑉)
9	1, 5, 6, 7, 8	setchom 18078	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑦 ↑m 𝑥))
10		resssetc.c	. . . . . 6 ⊢ 𝐶 = (SetCat‘𝑈)
11	2	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑈 ∈ 𝑊)
12		eqid 2728	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
13	3	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑉 ⊆ 𝑈)
14	13, 7	sseldd 3983	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝑈)
15	13, 8	sseldd 3983	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ 𝑈)
16	10, 11, 12, 14, 15	setchom 18078	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑦 ↑m 𝑥))
17		eqid 2728	. . . . . . . 8 ⊢ (𝐶 ↾s 𝑉) = (𝐶 ↾s 𝑉)
18	17, 12	resshom 17409	. . . . . . 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 7454	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦))
21	9, 16, 20	3eqtr2rd 2775	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
22	21	ralrimivva 3198	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
23		eqid 2728	. . . 4 ⊢ (Hom ‘(𝐶 ↾s 𝑉)) = (Hom ‘(𝐶 ↾s 𝑉))
24	10, 2	setcbas 18076	. . . . . 6 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
25	3, 24	sseqtrd 4022	. . . . 5 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐶))
26		eqid 2728	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
27	17, 26	ressbas2 17227	. . . . 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 18076	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝐷))
30	23, 6, 28, 29	homfeq 17683	. . 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 256	. 2 ⊢ (𝜑 → (Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷))
32	4	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉 ∈ V)
33		eqid 2728	. . . . . . . 8 ⊢ (comp‘𝐷) = (comp‘𝐷)
34		simplr1 1212	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥 ∈ 𝑉)
35		simplr2 1213	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦 ∈ 𝑉)
36		simplr3 1214	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧 ∈ 𝑉)
37		simprl 769	. . . . . . . . 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 18079	. . . . . . . . 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 231	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓:𝑥⟶𝑦)
40		simprr 771	. . . . . . . . 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 18079	. . . . . . . . 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 231	. . . . . . . 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 18081	. . . . . . 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 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑈 ∈ 𝑊)
45		eqid 2728	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
46	3	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉 ⊆ 𝑈)
47	46, 34	sseldd 3983	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥 ∈ 𝑈)
48	46, 35	sseldd 3983	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦 ∈ 𝑈)
49	46, 36	sseldd 3983	. . . . . . . 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 18081	. . . . . . 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 17410	. . . . . . . . . . 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 724	. . . . . . . . 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 7443	. . . . . . . 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 7443	. . . . . . 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 2774	. . . . . 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 3198	. . . . 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 3201	. . . 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 2728	. . . . 5 ⊢ (comp‘(𝐶 ↾s 𝑉)) = (comp‘(𝐶 ↾s 𝑉))
60	31	eqcomd 2734	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾s 𝑉)))
61	33, 59, 6, 29, 28, 60	comfeq 17695	. . . 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 256	. . 3 ⊢ (𝜑 → (compf‘𝐷) = (compf‘(𝐶 ↾s 𝑉)))
63	62	eqcomd 2734	. 2 ⊢ (𝜑 → (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷))
64	31, 63	jca 510	1 ⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ∧ (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷)))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3058  Vcvv 3473   ⊆ wss 3949  ⟨cop 4638   ∘ ccom 5686  ⟶wf 6549  ‘cfv 6553  (class class class)co 7426   ↑_m cmap 8853  Basecbs 17189   ↾_s cress 17218  Hom chom 17253  compcco 17254  Hom_f chomf 17655  comp_fccomf 17656  SetCatcsetc 18073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-er 8733  df-map 8855  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-nn 12253  df-2 12315  df-3 12316  df-4 12317  df-5 12318  df-6 12319  df-7 12320  df-8 12321  df-9 12322  df-n0 12513  df-z 12599  df-dec 12718  df-uz 12863  df-fz 13527  df-struct 17125  df-sets 17142  df-slot 17160  df-ndx 17172  df-base 17190  df-ress 17219  df-hom 17266  df-cco 17267  df-homf 17659  df-comf 17660  df-setc 18074

This theorem is referenced by:  funcsetcres2  18091