Theorem issubc3 17121

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 17971, 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 487	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → 𝐽 ∈ (Subcat‘𝐶))
2		issubc3.h	. . . 4 ⊢ 𝐻 = (Homf ‘𝐶)
3	1, 2	subcssc 17112	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → 𝐽 ⊆cat 𝐻)
4	1	adantr 483	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝐽 ∈ (Subcat‘𝐶))
5		issubc3.a	. . . . . 6 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
6	5	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝐽 Fn (𝑆 × 𝑆))
7		simpr 487	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
8		issubc3.i	. . . . 5 ⊢ 1 = (Id‘𝐶)
9	4, 6, 7, 8	subcidcl 17116	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥 ∈ 𝑆) → ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
10	9	ralrimiva 3184	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
11		issubc3.1	. . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐽)
12	11, 1	subccat 17120	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → 𝐷 ∈ Cat)
13	3, 10, 12	3jca 1124	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))
14		simpr1 1190	. . 3 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 ⊆cat 𝐻)
15		simpr2 1191	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
16		eqid 2823	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
17		eqid 2823	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18		eqid 2823	. . . . . . . . . 10 ⊢ (comp‘𝐷) = (comp‘𝐷)
19		simplrr 776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐷 ∈ Cat)
20		simprl1 1214	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥 ∈ 𝑆)
21		eqid 2823	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
22		issubc3.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ Cat)
23	22	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐶 ∈ Cat)
24	5	ad2antrr 724	. . . . . . . . . . . 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 16965	. . . . . . . . . . . . . 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 775	. . . . . . . . . . . . 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 17093	. . . . . . . . . . . 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 17101	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑆 = (Base‘𝐷))
30	20, 29	eleqtrd 2917	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥 ∈ (Base‘𝐷))
31		simprl2 1215	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦 ∈ 𝑆)
32	31, 29	eleqtrd 2917	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦 ∈ (Base‘𝐷))
33		simprl3 1216	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧 ∈ 𝑆)
34	33, 29	eleqtrd 2917	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧 ∈ (Base‘𝐷))
35		simprrl 779	. . . . . . . . . . 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 17102	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽 = (Hom ‘𝐷))
37	36	oveqd 7175	. . . . . . . . . . 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 2917	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
39		simprrr 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑔 ∈ (𝑦𝐽𝑧))
40	36	oveqd 7175	. . . . . . . . . . 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 2917	. . . . . . . . . 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 16958	. . . . . . . . 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 2823	. . . . . . . . . . . 12 ⊢ (comp‘𝐶) = (comp‘𝐶)
44	11, 21, 23, 24, 28, 43	rescco 17104	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (comp‘𝐶) = (comp‘𝐷))
45	44	oveqd 7175	. . . . . . . . . 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 7175	. . . . . . . . 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 7175	. . . . . . . . 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 2930	. . . . . . . 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 470	. . . . . . 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 3193	. . . . . 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 3194	. . . . 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 1166	. . . 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 3172	. . . 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 585	. . 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 483	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐶 ∈ Cat)
56	5	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 Fn (𝑆 × 𝑆))
57	2, 8, 43, 55, 56	issubc2 17108	. . 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 711	. 2 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 ∈ (Subcat‘𝐶))
59	13, 58	impbida 799	1 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ⟨cop 4575   class class class wbr 5068   × cxp 5555   Fn wfn 6352  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  Hom chom 16578  compcco 16579  Catccat 16937  Idccid 16938  Hom_f chomf 16939   ⊆_cat cssc 17079   ↾_cat cresc 17080  Subcatcsubc 17081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-pm 8411  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-hom 16591  df-cco 16592  df-cat 16941  df-cid 16942  df-homf 16943  df-ssc 17082  df-resc 17083  df-subc 17084

This theorem is referenced by:  subsubc  17125