Theorem issubc3 17480

Description: Alternate definition of a subcategory, as a subset of the category which is itself a category. The assumption that the identity be closed is necessary just as in the case of a monoid, issubm2 18358, for the same reasons, since categories are a generalization of monoids. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
issubc3.h	⊢ 𝐻 = (Homf ‘𝐶)
issubc3.i	⊢ 1 = (Id‘𝐶)
issubc3.1	⊢ 𝐷 = (𝐶 ↾cat 𝐽)
issubc3.c	⊢ (𝜑 → 𝐶 ∈ Cat)
issubc3.a	⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))

Assertion

Ref	Expression
issubc3	⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)))

Distinct variable groups:   𝑥,𝐶   𝑥,𝐷   𝑥,𝐻   𝜑,𝑥   𝑥,𝐽   𝑥,𝑆

Allowed substitution hint:   1 (𝑥)

Proof of Theorem issubc3

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

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → 𝐽 ∈ (Subcat‘𝐶))
2		issubc3.h	. . . 4 ⊢ 𝐻 = (Homf ‘𝐶)
3	1, 2	subcssc 17471	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → 𝐽 ⊆cat 𝐻)
4	1	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝐽 ∈ (Subcat‘𝐶))
5		issubc3.a	. . . . . 6 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
6	5	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝐽 Fn (𝑆 × 𝑆))
7		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
8		issubc3.i	. . . . 5 ⊢ 1 = (Id‘𝐶)
9	4, 6, 7, 8	subcidcl 17475	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥 ∈ 𝑆) → ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
10	9	ralrimiva 3107	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
11		issubc3.1	. . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐽)
12	11, 1	subccat 17479	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → 𝐷 ∈ Cat)
13	3, 10, 12	3jca 1126	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))
14		simpr1 1192	. . 3 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 ⊆cat 𝐻)
15		simpr2 1193	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
16		eqid 2738	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
17		eqid 2738	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18		eqid 2738	. . . . . . . . . 10 ⊢ (comp‘𝐷) = (comp‘𝐷)
19		simplrr 774	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐷 ∈ Cat)
20		simprl1 1216	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥 ∈ 𝑆)
21		eqid 2738	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
22		issubc3.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ Cat)
23	22	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐶 ∈ Cat)
24	5	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽 Fn (𝑆 × 𝑆))
25	2, 21	homffn 17319	. . . . . . . . . . . . . 14 ⊢ 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶))
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)))
27		simplrl 773	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽 ⊆cat 𝐻)
28	24, 26, 27	ssc1 17450	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑆 ⊆ (Base‘𝐶))
29	11, 21, 23, 24, 28	rescbas 17458	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑆 = (Base‘𝐷))
30	20, 29	eleqtrd 2841	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥 ∈ (Base‘𝐷))
31		simprl2 1217	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦 ∈ 𝑆)
32	31, 29	eleqtrd 2841	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦 ∈ (Base‘𝐷))
33		simprl3 1218	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧 ∈ 𝑆)
34	33, 29	eleqtrd 2841	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧 ∈ (Base‘𝐷))
35		simprrl 777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑓 ∈ (𝑥𝐽𝑦))
36	11, 21, 23, 24, 28	reschom 17460	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽 = (Hom ‘𝐷))
37	36	oveqd 7272	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑥𝐽𝑦) = (𝑥(Hom ‘𝐷)𝑦))
38	35, 37	eleqtrd 2841	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
39		simprrr 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑔 ∈ (𝑦𝐽𝑧))
40	36	oveqd 7272	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑦𝐽𝑧) = (𝑦(Hom ‘𝐷)𝑧))
41	39, 40	eleqtrd 2841	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
42	16, 17, 18, 19, 30, 32, 34, 38, 41	catcocl 17311	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧))
43		eqid 2738	. . . . . . . . . . . 12 ⊢ (comp‘𝐶) = (comp‘𝐶)
44	11, 21, 23, 24, 28, 43	rescco 17462	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (comp‘𝐶) = (comp‘𝐷))
45	44	oveqd 7272	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧) = (⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧))
46	45	oveqd 7272	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))
47	36	oveqd 7272	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑥𝐽𝑧) = (𝑥(Hom ‘𝐷)𝑧))
48	42, 46, 47	3eltr4d 2854	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
49	48	anassrs 467	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
50	49	ralrimivva 3114	. . . . . 6 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
51	50	ralrimivvva 3115	. . . . 5 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
52	51	3adantr2 1168	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
53		r19.26 3094	. . . 4 ⊢ (∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) ↔ (∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
54	15, 52, 53	sylanbrc 582	. . 3 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
55	22	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐶 ∈ Cat)
56	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 Fn (𝑆 × 𝑆))
57	2, 8, 43, 55, 56	issubc2 17467	. . 3 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
58	14, 54, 57	mpbir2and 709	. 2 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 ∈ (Subcat‘𝐶))
59	13, 58	impbida 797	1 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ⟨cop 4564   class class class wbr 5070   × cxp 5578   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  compcco 16900  Catccat 17290  Idccid 17291  Hom_f chomf 17292   ⊆_cat cssc 17436   ↾_cat cresc 17437  Subcatcsubc 17438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-hom 16912  df-cco 16913  df-cat 17294  df-cid 17295  df-homf 17296  df-ssc 17439  df-resc 17440  df-subc 17441

This theorem is referenced by:  subsubc  17484