Theorem issubc3 17797

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 18718, 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 17788	. . 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 723	. . . . 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 17792	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥 ∈ 𝑆) → ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
10	9	ralrimiva 3138	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
11		issubc3.1	. . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐽)
12	11, 1	subccat 17796	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → 𝐷 ∈ Cat)
13	3, 10, 12	3jca 1125	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))
14		simpr1 1191	. . 3 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 ⊆cat 𝐻)
15		simpr2 1192	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
16		eqid 2724	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
17		eqid 2724	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18		eqid 2724	. . . . . . . . . 10 ⊢ (comp‘𝐷) = (comp‘𝐷)
19		simplrr 775	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐷 ∈ Cat)
20		simprl1 1215	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥 ∈ 𝑆)
21		eqid 2724	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
22		issubc3.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ Cat)
23	22	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐶 ∈ Cat)
24	5	ad2antrr 723	. . . . . . . . . . . 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 17635	. . . . . . . . . . . . . 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 774	. . . . . . . . . . . . 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 17766	. . . . . . . . . . . 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 17774	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑆 = (Base‘𝐷))
30	20, 29	eleqtrd 2827	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥 ∈ (Base‘𝐷))
31		simprl2 1216	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦 ∈ 𝑆)
32	31, 29	eleqtrd 2827	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦 ∈ (Base‘𝐷))
33		simprl3 1217	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧 ∈ 𝑆)
34	33, 29	eleqtrd 2827	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧 ∈ (Base‘𝐷))
35		simprrl 778	. . . . . . . . . . 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 17776	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽 = (Hom ‘𝐷))
37	36	oveqd 7418	. . . . . . . . . . 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 2827	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
39		simprrr 779	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑔 ∈ (𝑦𝐽𝑧))
40	36	oveqd 7418	. . . . . . . . . . 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 2827	. . . . . . . . . 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 17627	. . . . . . . . 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 2724	. . . . . . . . . . . 12 ⊢ (comp‘𝐶) = (comp‘𝐶)
44	11, 21, 23, 24, 28, 43	rescco 17778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (comp‘𝐶) = (comp‘𝐷))
45	44	oveqd 7418	. . . . . . . . . 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 7418	. . . . . . . . 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 7418	. . . . . . . . 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 2840	. . . . . . . 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 3192	. . . . . 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 3195	. . . . 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 1167	. . . 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 3103	. . . 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 17784	. . 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 710	. 2 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 ∈ (Subcat‘𝐶))
59	13, 58	impbida 798	1 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3053  ⟨cop 4626   class class class wbr 5138   × cxp 5664   Fn wfn 6528  ‘cfv 6533  (class class class)co 7401  Basecbs 17142  Hom chom 17206  compcco 17207  Catccat 17606  Idccid 17607  Hom_f chomf 17608   ⊆_cat cssc 17752   ↾_cat cresc 17753  Subcatcsubc 17754

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 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8698  df-pm 8818  df-ixp 8887  df-en 8935  df-dom 8936  df-sdom 8937  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17143  df-ress 17172  df-hom 17219  df-cco 17220  df-cat 17610  df-cid 17611  df-homf 17612  df-ssc 17755  df-resc 17756  df-subc 17757

This theorem is referenced by:  subsubc  17801