Theorem fullsubc 17775

Description: The full subcategory generated by a subset of objects is the category with these objects and the same morphisms as the original. The result is always a subcategory (and it is full, meaning that all morphisms of the original category between objects in the subcategory is also in the subcategory), see definition 4.1(2) of [Adamek] p. 48. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
fullsubc.b	⊢ 𝐵 = (Base‘𝐶)
fullsubc.h	⊢ 𝐻 = (Homf ‘𝐶)
fullsubc.c	⊢ (𝜑 → 𝐶 ∈ Cat)
fullsubc.s	⊢ (𝜑 → 𝑆 ⊆ 𝐵)

Assertion

Ref	Expression
fullsubc	⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐶))

Proof of Theorem fullsubc

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

Step	Hyp	Ref	Expression
1		fullsubc.h	. . . . 5 ⊢ 𝐻 = (Homf ‘𝐶)
2		fullsubc.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
3	1, 2	homffn 17617	. . . 4 ⊢ 𝐻 Fn (𝐵 × 𝐵)
4	2	fvexi 6840	. . . 4 ⊢ 𝐵 ∈ V
5		sscres 17748	. . . 4 ⊢ ((𝐻 Fn (𝐵 × 𝐵) ∧ 𝐵 ∈ V) → (𝐻 ↾ (𝑆 × 𝑆)) ⊆cat 𝐻)
6	3, 4, 5	mp2an 692	. . 3 ⊢ (𝐻 ↾ (𝑆 × 𝑆)) ⊆cat 𝐻
7	6	a1i 11	. 2 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) ⊆cat 𝐻)
8		eqid 2729	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
9		eqid 2729	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
10		fullsubc.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐶 ∈ Cat)
12		fullsubc.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
13	12	sselda 3937	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
14	2, 8, 9, 11, 13	catidcl 17606	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
15		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
16	15, 15	ovresd 7520	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) = (𝑥𝐻𝑥))
17	1, 2, 8, 13, 13	homfval 17616	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝐻𝑥) = (𝑥(Hom ‘𝐶)𝑥))
18	16, 17	eqtrd 2764	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) = (𝑥(Hom ‘𝐶)𝑥))
19	14, 18	eleqtrrd 2831	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥))
20		eqid 2729	. . . . . . . . . 10 ⊢ (comp‘𝐶) = (comp‘𝐶)
21	11	ad3antrrr 730	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐶 ∈ Cat)
22	13	ad3antrrr 730	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥 ∈ 𝐵)
23	12	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ 𝐵)
24	23	sselda 3937	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝐵)
25	24	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) → 𝑦 ∈ 𝐵)
26	25	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦 ∈ 𝐵)
27	23	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) → 𝑆 ⊆ 𝐵)
28	27	sselda 3937	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝐵)
29	28	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ 𝐵)
30		simprl 770	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
31		simprr 772	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
32	2, 8, 20, 21, 22, 26, 29, 30, 31	catcocl 17609	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
33	15	ad3antrrr 730	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥 ∈ 𝑆)
34		simplr 768	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ 𝑆)
35	33, 34	ovresd 7520	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑥𝐻𝑧))
36	1, 2, 8, 22, 29	homfval 17616	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥𝐻𝑧) = (𝑥(Hom ‘𝐶)𝑧))
37	35, 36	eqtrd 2764	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑥(Hom ‘𝐶)𝑧))
38	32, 37	eleqtrrd 2831	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧))
39	38	ralrimivva 3172	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧))
40		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆)
41		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆)
42	40, 41	ovresd 7520	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥𝐻𝑦))
43	13	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝐵)
44	1, 2, 8, 43, 24	homfval 17616	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
45	42, 44	eqtrd 2764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐶)𝑦))
46	45	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐶)𝑦))
47		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) → 𝑦 ∈ 𝑆)
48		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑆)
49	47, 48	ovresd 7520	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) → (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑦𝐻𝑧))
50	1, 2, 8, 25, 28	homfval 17616	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) → (𝑦𝐻𝑧) = (𝑦(Hom ‘𝐶)𝑧))
51	49, 50	eqtrd 2764	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) → (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑦(Hom ‘𝐶)𝑧))
52	51	raleqdv 3290	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) → (∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧) ↔ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧)))
53	46, 52	raleqbidv 3310	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) → (∀𝑓 ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦)∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧)))
54	39, 53	mpbird 257	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑆) → ∀𝑓 ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦)∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧))
55	54	ralrimiva 3121	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) → ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦)∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧))
56	55	ralrimiva 3121	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦)∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧))
57	19, 56	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (((Id‘𝐶)‘𝑥) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦)∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧)))
58	57	ralrimiva 3121	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (((Id‘𝐶)‘𝑥) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦)∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧)))
59		xpss12 5638	. . . . 5 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
60	12, 12, 59	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
61		fnssres 6609	. . . 4 ⊢ ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
62	3, 60, 61	sylancr 587	. . 3 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
63	1, 9, 20, 10, 62	issubc2 17761	. 2 ⊢ (𝜑 → ((𝐻 ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐶) ↔ ((𝐻 ↾ (𝑆 × 𝑆)) ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (((Id‘𝐶)‘𝑥) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦)∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧)))))
64	7, 58, 63	mpbir2and 713	1 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3438   ⊆ wss 3905  ⟨cop 4585   class class class wbr 5095   × cxp 5621   ↾ cres 5625   Fn wfn 6481  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  Hom chom 17190  compcco 17191  Catccat 17588  Idccid 17589  Hom_f chomf 17590   ⊆_cat cssc 17732  Subcatcsubc 17734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-pm 8763  df-ixp 8832  df-cat 17592  df-cid 17593  df-homf 17594  df-ssc 17735  df-subc 17737

This theorem is referenced by:  resscat  17777  funcres2c  17828  ressffth  17865  funcsetcres2  18018  imasubc2  49125