Theorem issubc3 18017

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 18992, 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 490	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → 𝐽 ∈ (Subcat‘𝐶))
2		issubc3.h	. . . 4 ⊢ 𝐻 = (Homf ‘𝐶)
3	1, 2	subcssc 18008	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → 𝐽 ⊆cat 𝐻)
4	1	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝐽 ∈ (Subcat‘𝐶))
5		issubc3.a	. . . . . 6 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
6	5	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝐽 Fn (𝑆 × 𝑆))
7		simpr 490	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
8		issubc3.i	. . . . 5 ⊢ 1 = (Id‘𝐶)
9	4, 6, 7, 8	subcidcl 18012	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥 ∈ 𝑆) → ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
10	9	ralrimiva 3155	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
11		issubc3.1	. . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐽)
12	11, 1	subccat 18016	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → 𝐷 ∈ Cat)
13	3, 10, 12	3jca 1146	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ (Subcat‘𝐶)) → (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))
14		simpr1 1213	. . 3 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 ⊆cat 𝐻)
15		simpr2 1214	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
16		eqid 2761	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
17		eqid 2761	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18		eqid 2761	. . . . . . . . . 10 ⊢ (comp‘𝐷) = (comp‘𝐷)
19		simplrr 790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐷 ∈ Cat)
20		simprl1 1237	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥 ∈ 𝑆)
21		eqid 2761	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
22		issubc3.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ Cat)
23	22	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐶 ∈ Cat)
24	5	ad2antrr 739	. . . . . . . . . . . 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 17860	. . . . . . . . . . . . . 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 789	. . . . . . . . . . . . 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 17989	. . . . . . . . . . . 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 17997	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑆 = (Base‘𝐷))
30	20, 29	eleqtrd 2863	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥 ∈ (Base‘𝐷))
31		simprl2 1238	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦 ∈ 𝑆)
32	31, 29	eleqtrd 2863	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦 ∈ (Base‘𝐷))
33		simprl3 1239	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧 ∈ 𝑆)
34	33, 29	eleqtrd 2863	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧 ∈ (Base‘𝐷))
35		simprrl 793	. . . . . . . . . . 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 17998	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽 = (Hom ‘𝐷))
37	36	oveqd 7435	. . . . . . . . . . 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 2863	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
39		simprrr 794	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑔 ∈ (𝑦𝐽𝑧))
40	36	oveqd 7435	. . . . . . . . . . 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 2863	. . . . . . . . . 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 17852	. . . . . . . . 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 2761	. . . . . . . . . . . 12 ⊢ (comp‘𝐶) = (comp‘𝐶)
44	11, 21, 23, 24, 28, 43	rescco 18000	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ 𝐷 ∈ Cat)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (comp‘𝐶) = (comp‘𝐷))
45	44	oveqd 7435	. . . . . . . . . 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 7435	. . . . . . . . 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 7435	. . . . . . . . 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 2876	. . . . . . . 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 473	. . . . . . 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 3206	. . . . . 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 3209	. . . . 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 1189	. . . 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 3123	. . . 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 595	. . 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 486	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐶 ∈ Cat)
56	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 Fn (𝑆 × 𝑆))
57	2, 8, 43, 55, 56	issubc2 18004	. . 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 726	. 2 ⊢ ((𝜑 ∧ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 ∈ (Subcat‘𝐶))
59	13, 58	impbida 813	1 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ⟨cop 4590   class class class wbr 5103   × cxp 5649   Fn wfn 6532  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  Hom chom 17432  compcco 17433  Catccat 17831  Idccid 17832  Hom_f chomf 17833   ⊆_cat cssc 17975   ↾_cat cresc 17976  Subcatcsubc 17977

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-er 8710  df-pm 8843  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-hom 17445  df-cco 17446  df-cat 17835  df-cid 17836  df-homf 17837  df-ssc 17978  df-resc 17979  df-subc 17980

This theorem is used by:  subsubc  18021