Theorem issubc3 17764

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 18720, 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 17755	. . 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 726	. . . . 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 17759	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥 ∈ 𝑆) → ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
10	9	ralrimiva 3125	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
11		issubc3.1	. . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐽)
12	11, 1	subccat 17763	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → 𝐷 ∈ Cat)
13	3, 10, 12	3jca 1128	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))
14		simpr1 1195	. . 3 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 ⊆cat 𝐻)
15		simpr2 1196	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
16		eqid 2733	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
17		eqid 2733	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18		eqid 2733	. . . . . . . . . 10 ⊢ (comp‘𝐷) = (comp‘𝐷)
19		simplrr 777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐷 ∈ Cat)
20		simprl1 1219	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥 ∈ 𝑆)
21		eqid 2733	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
22		issubc3.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ Cat)
23	22	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐶 ∈ Cat)
24	5	ad2antrr 726	. . . . . . . . . . . 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 17607	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . 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 17736	. . . . . . . . . . . 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 17744	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑆 = (Base‘𝐷))
30	20, 29	eleqtrd 2835	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥 ∈ (Base‘𝐷))
31		simprl2 1220	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦 ∈ 𝑆)
32	31, 29	eleqtrd 2835	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦 ∈ (Base‘𝐷))
33		simprl3 1221	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧 ∈ 𝑆)
34	33, 29	eleqtrd 2835	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧 ∈ (Base‘𝐷))
35		simprrl 780	. . . . . . . . . . 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 17745	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽 = (Hom ‘𝐷))
37	36	oveqd 7372	. . . . . . . . . . 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 2835	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
39		simprrr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑔 ∈ (𝑦𝐽𝑧))
40	36	oveqd 7372	. . . . . . . . . . 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 2835	. . . . . . . . . 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 17599	. . . . . . . . 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 2733	. . . . . . . . . . . 12 ⊢ (comp‘𝐶) = (comp‘𝐶)
44	11, 21, 23, 24, 28, 43	rescco 17747	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (comp‘𝐶) = (comp‘𝐷))
45	44	oveqd 7372	. . . . . . . . . 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 7372	. . . . . . . . 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 7372	. . . . . . . . 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 2848	. . . . . . . 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 3176	. . . . . 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 3179	. . . . 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 1171	. . . 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 3093	. . . 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 583	. . 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 17751	. . 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 713	. 2 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 ∈ (Subcat‘𝐶))
59	13, 58	impbida 800	1 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ⟨cop 4583   class class class wbr 5095   × cxp 5619   Fn wfn 6484  ‘cfv 6489  (class class class)co 7355  Basecbs 17127  Hom chom 17179  compcco 17180  Catccat 17578  Idccid 17579  Hom_f chomf 17580   ⊆_cat cssc 17722   ↾_cat cresc 17723  Subcatcsubc 17724

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-er 8631  df-pm 8762  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-2 12199  df-3 12200  df-4 12201  df-5 12202  df-6 12203  df-7 12204  df-8 12205  df-9 12206  df-n0 12393  df-z 12480  df-dec 12599  df-sets 17082  df-slot 17100  df-ndx 17112  df-base 17128  df-ress 17149  df-hom 17192  df-cco 17193  df-cat 17582  df-cid 17583  df-homf 17584  df-ssc 17725  df-resc 17726  df-subc 17727

This theorem is referenced by:  subsubc  17768