Theorem discsubc 50317

Description: A discrete category, whose only morphisms are the identity morphisms, is a subcategory. (Contributed by Zhi Wang, 1-Nov-2025.)

Hypotheses

Ref	Expression
discsubc.j	⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
discsubc.b	⊢ 𝐵 = (Base‘𝐶)
discsubc.i	⊢ 𝐼 = (Id‘𝐶)
discsubc.s	⊢ (𝜑 → 𝑆 ⊆ 𝐵)
discsubc.c	⊢ (𝜑 → 𝐶 ∈ Cat)

Assertion

Ref	Expression
discsubc	⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))

Distinct variable groups:   𝑥,𝑆,𝑦   𝑥,𝐼,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐵(𝑥, 𝑦)   𝐶(𝑥, 𝑦)   𝐽(𝑥, 𝑦)

Proof of Theorem discsubc

Dummy variables 𝑎 𝑏 𝑐 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		discsubc.s	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
2		eqeq12 2780	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥 = 𝑦 ↔ 𝑎 = 𝑏))
3		simpl 488	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑥 = 𝑎)
4	3	fveq2d 6897	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝐼‘𝑥) = (𝐼‘𝑎))
5	4	sneqd 4595	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {(𝐼‘𝑥)} = {(𝐼‘𝑎)})
6	2, 5	ifbieq1d 4509	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) = if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅))
7		discsubc.j	. . . . . . 7 ⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
8		snex 5400	. . . . . . . 8 ⊢ {(𝐼‘𝑎)} ∈ V
9		0ex 5263	. . . . . . . 8 ⊢ ∅ ∈ V
10	8, 9	ifex 4535	. . . . . . 7 ⊢ if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅) ∈ V
11	6, 7, 10	ovmpoa 7582	. . . . . 6 ⊢ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) → (𝑎𝐽𝑏) = if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅))
12	11	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎𝐽𝑏) = if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅))
13		sseq1 3937	. . . . . 6 ⊢ ({(𝐼‘𝑎)} = if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅) → ({(𝐼‘𝑎)} ⊆ (𝑎(Homf ‘𝐶)𝑏) ↔ if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅) ⊆ (𝑎(Homf ‘𝐶)𝑏)))
14		sseq1 3937	. . . . . 6 ⊢ (∅ = if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅) → (∅ ⊆ (𝑎(Homf ‘𝐶)𝑏) ↔ if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅) ⊆ (𝑎(Homf ‘𝐶)𝑏)))
15		discsubc.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
16		eqid 2763	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
17		discsubc.i	. . . . . . . . 9 ⊢ 𝐼 = (Id‘𝐶)
18		discsubc.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ Cat)
19	18	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) ∧ 𝑎 = 𝑏) → 𝐶 ∈ Cat)
20	1	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) ∧ 𝑎 = 𝑏) → 𝑆 ⊆ 𝐵)
21		simplrl 789	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) ∧ 𝑎 = 𝑏) → 𝑎 ∈ 𝑆)
22	20, 21	sseldd 3913	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) ∧ 𝑎 = 𝑏) → 𝑎 ∈ 𝐵)
23	15, 16, 17, 19, 22	catidcl 17868	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) ∧ 𝑎 = 𝑏) → (𝐼‘𝑎) ∈ (𝑎(Hom ‘𝐶)𝑎))
24		eqid 2763	. . . . . . . . . 10 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
25	24, 15, 16, 22, 22	homfval 17878	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) ∧ 𝑎 = 𝑏) → (𝑎(Homf ‘𝐶)𝑎) = (𝑎(Hom ‘𝐶)𝑎))
26		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) ∧ 𝑎 = 𝑏) → 𝑎 = 𝑏)
27	26	oveq2d 7443	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) ∧ 𝑎 = 𝑏) → (𝑎(Homf ‘𝐶)𝑎) = (𝑎(Homf ‘𝐶)𝑏))
28	25, 27	eqtr3d 2800	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) ∧ 𝑎 = 𝑏) → (𝑎(Hom ‘𝐶)𝑎) = (𝑎(Homf ‘𝐶)𝑏))
29	23, 28	eleqtrd 2865	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) ∧ 𝑎 = 𝑏) → (𝐼‘𝑎) ∈ (𝑎(Homf ‘𝐶)𝑏))
30	29	snssd 4746	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) ∧ 𝑎 = 𝑏) → {(𝐼‘𝑎)} ⊆ (𝑎(Homf ‘𝐶)𝑏))
31		0ss 4352	. . . . . . 7 ⊢ ∅ ⊆ (𝑎(Homf ‘𝐶)𝑏)
32	31	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) ∧ ¬ 𝑎 = 𝑏) → ∅ ⊆ (𝑎(Homf ‘𝐶)𝑏))
33	13, 14, 30, 32	ifbothda 4523	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅) ⊆ (𝑎(Homf ‘𝐶)𝑏))
34	12, 33	eqsstrd 3946	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎𝐽𝑏) ⊆ (𝑎(Homf ‘𝐶)𝑏))
35	34	ralrimivva 3208	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝐽𝑏) ⊆ (𝑎(Homf ‘𝐶)𝑏))
36	7	discsubclem 50316	. . . . 5 ⊢ 𝐽 Fn (𝑆 × 𝑆)
37	36	a1i 11	. . . 4 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
38	24, 15	homffn 17879	. . . . 5 ⊢ (Homf ‘𝐶) Fn (𝐵 × 𝐵)
39	38	a1i 11	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) Fn (𝐵 × 𝐵))
40	15	fvexi 6907	. . . . 5 ⊢ 𝐵 ∈ V
41	40	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
42	37, 39, 41	isssc 18007	. . 3 ⊢ (𝜑 → (𝐽 ⊆cat (Homf ‘𝐶) ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝐽𝑏) ⊆ (𝑎(Homf ‘𝐶)𝑏))))
43	1, 35, 42	mpbir2and 726	. 2 ⊢ (𝜑 → 𝐽 ⊆cat (Homf ‘𝐶))
44		fvex 6906	. . . . . 6 ⊢ (𝐼‘𝑎) ∈ V
45	44	snid 4622	. . . . 5 ⊢ (𝐼‘𝑎) ∈ {(𝐼‘𝑎)}
46		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
47		equtr2 2061	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → 𝑥 = 𝑦)
48	47	iftrued 4492	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) = {(𝐼‘𝑥)})
49		simpl 488	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → 𝑥 = 𝑎)
50	49	fveq2d 6897	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐼‘𝑥) = (𝐼‘𝑎))
51	50	sneqd 4595	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → {(𝐼‘𝑥)} = {(𝐼‘𝑎)})
52	48, 51	eqtrd 2798	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) = {(𝐼‘𝑎)})
53	52, 7, 8	ovmpoa 7582	. . . . . 6 ⊢ ((𝑎 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆) → (𝑎𝐽𝑎) = {(𝐼‘𝑎)})
54	46, 46, 53	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎𝐽𝑎) = {(𝐼‘𝑎)})
55	45, 54	eleqtrrid 2870	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝐼‘𝑎) ∈ (𝑎𝐽𝑎))
56	45	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → (𝐼‘𝑎) ∈ {(𝐼‘𝑎)})
57		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑓 ∈ (𝑎𝐽𝑏))
58	46	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑎 ∈ 𝑆)
59		simplrl 789	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑏 ∈ 𝑆)
60	58, 59, 11	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → (𝑎𝐽𝑏) = if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅))
61	57, 60	eleqtrd 2865	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑓 ∈ if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅))
62	61	ne0d 4290	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅) ≠ ∅)
63		iffalse 4493	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑎 = 𝑏 → if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅) = ∅)
64	63	necon1ai 2985	. . . . . . . . . . . . 13 ⊢ (if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅) ≠ ∅ → 𝑎 = 𝑏)
65	62, 64	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑎 = 𝑏)
66	65	opeq2d 4849	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → ⟨𝑎, 𝑎⟩ = ⟨𝑎, 𝑏⟩)
67		simprr 785	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑔 ∈ (𝑏𝐽𝑐))
68		eqeq12 2780	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑐) → (𝑥 = 𝑦 ↔ 𝑏 = 𝑐))
69		simpl 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑐) → 𝑥 = 𝑏)
70	69	fveq2d 6897	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑐) → (𝐼‘𝑥) = (𝐼‘𝑏))
71	70	sneqd 4595	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑐) → {(𝐼‘𝑥)} = {(𝐼‘𝑏)})
72	68, 71	ifbieq1d 4509	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑐) → if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) = if(𝑏 = 𝑐, {(𝐼‘𝑏)}, ∅))
73		snex 5400	. . . . . . . . . . . . . . . . . 18 ⊢ {(𝐼‘𝑏)} ∈ V
74	73, 9	ifex 4535	. . . . . . . . . . . . . . . . 17 ⊢ if(𝑏 = 𝑐, {(𝐼‘𝑏)}, ∅) ∈ V
75	72, 7, 74	ovmpoa 7582	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆) → (𝑏𝐽𝑐) = if(𝑏 = 𝑐, {(𝐼‘𝑏)}, ∅))
76	75	ad2antlr 740	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → (𝑏𝐽𝑐) = if(𝑏 = 𝑐, {(𝐼‘𝑏)}, ∅))
77	67, 76	eleqtrd 2865	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑔 ∈ if(𝑏 = 𝑐, {(𝐼‘𝑏)}, ∅))
78	77	ne0d 4290	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → if(𝑏 = 𝑐, {(𝐼‘𝑏)}, ∅) ≠ ∅)
79		iffalse 4493	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑏 = 𝑐 → if(𝑏 = 𝑐, {(𝐼‘𝑏)}, ∅) = ∅)
80	79	necon1ai 2985	. . . . . . . . . . . . 13 ⊢ (if(𝑏 = 𝑐, {(𝐼‘𝑏)}, ∅) ≠ ∅ → 𝑏 = 𝑐)
81	78, 80	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑏 = 𝑐)
82	65, 81	eqtrd 2798	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑎 = 𝑐)
83	66, 82	oveq12d 7445	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → (⟨𝑎, 𝑎⟩(comp‘𝐶)𝑎) = (⟨𝑎, 𝑏⟩(comp‘𝐶)𝑐))
84	83	eqcomd 2769	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → (⟨𝑎, 𝑏⟩(comp‘𝐶)𝑐) = (⟨𝑎, 𝑎⟩(comp‘𝐶)𝑎))
85	81	iftrued 4492	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → if(𝑏 = 𝑐, {(𝐼‘𝑏)}, ∅) = {(𝐼‘𝑏)})
86	77, 85	eleqtrd 2865	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑔 ∈ {(𝐼‘𝑏)})
87	86	elsnd 4601	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑔 = (𝐼‘𝑏))
88	65	fveq2d 6897	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → (𝐼‘𝑎) = (𝐼‘𝑏))
89	87, 88	eqtr4d 2801	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑔 = (𝐼‘𝑎))
90	65	iftrued 4492	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → if(𝑎 = 𝑏, {(𝐼‘𝑎)}, ∅) = {(𝐼‘𝑎)})
91	61, 90	eleqtrd 2865	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑓 ∈ {(𝐼‘𝑎)})
92	91	elsnd 4601	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑓 = (𝐼‘𝑎))
93	84, 89, 92	oveq123d 7448	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → (𝑔(⟨𝑎, 𝑏⟩(comp‘𝐶)𝑐)𝑓) = ((𝐼‘𝑎)(⟨𝑎, 𝑎⟩(comp‘𝐶)𝑎)(𝐼‘𝑎)))
94	18	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝐶 ∈ Cat)
95	1	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑆 ⊆ 𝐵)
96	95, 58	sseldd 3913	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → 𝑎 ∈ 𝐵)
97		eqid 2763	. . . . . . . . 9 ⊢ (comp‘𝐶) = (comp‘𝐶)
98	15, 16, 17, 94, 96	catidcl 17868	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → (𝐼‘𝑎) ∈ (𝑎(Hom ‘𝐶)𝑎))
99	15, 16, 17, 94, 96, 97, 96, 98	catlid 17869	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → ((𝐼‘𝑎)(⟨𝑎, 𝑎⟩(comp‘𝐶)𝑎)(𝐼‘𝑎)) = (𝐼‘𝑎))
100	93, 99	eqtrd 2798	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → (𝑔(⟨𝑎, 𝑏⟩(comp‘𝐶)𝑐)𝑓) = (𝐼‘𝑎))
101	82	oveq2d 7443	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → (𝑎𝐽𝑎) = (𝑎𝐽𝑐))
102	58, 58, 53	syl2anc 596	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → (𝑎𝐽𝑎) = {(𝐼‘𝑎)})
103	101, 102	eqtr3d 2800	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → (𝑎𝐽𝑐) = {(𝐼‘𝑎)})
104	56, 100, 103	3eltr4d 2878	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐽𝑏) ∧ 𝑔 ∈ (𝑏𝐽𝑐))) → (𝑔(⟨𝑎, 𝑏⟩(comp‘𝐶)𝑐)𝑓) ∈ (𝑎𝐽𝑐))
105	104	ralrimivva 3208	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) → ∀𝑓 ∈ (𝑎𝐽𝑏)∀𝑔 ∈ (𝑏𝐽𝑐)(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐶)𝑐)𝑓) ∈ (𝑎𝐽𝑐))
106	105	ralrimivva 3208	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ∀𝑏 ∈ 𝑆 ∀𝑐 ∈ 𝑆 ∀𝑓 ∈ (𝑎𝐽𝑏)∀𝑔 ∈ (𝑏𝐽𝑐)(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐶)𝑐)𝑓) ∈ (𝑎𝐽𝑐))
107	55, 106	jca 521	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((𝐼‘𝑎) ∈ (𝑎𝐽𝑎) ∧ ∀𝑏 ∈ 𝑆 ∀𝑐 ∈ 𝑆 ∀𝑓 ∈ (𝑎𝐽𝑏)∀𝑔 ∈ (𝑏𝐽𝑐)(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐶)𝑐)𝑓) ∈ (𝑎𝐽𝑐)))
108	107	ralrimiva 3157	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 ((𝐼‘𝑎) ∈ (𝑎𝐽𝑎) ∧ ∀𝑏 ∈ 𝑆 ∀𝑐 ∈ 𝑆 ∀𝑓 ∈ (𝑎𝐽𝑏)∀𝑔 ∈ (𝑏𝐽𝑐)(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐶)𝑐)𝑓) ∈ (𝑎𝐽𝑐)))
109	24, 17, 97, 18, 37	issubc2 18023	. 2 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat (Homf ‘𝐶) ∧ ∀𝑎 ∈ 𝑆 ((𝐼‘𝑎) ∈ (𝑎𝐽𝑎) ∧ ∀𝑏 ∈ 𝑆 ∀𝑐 ∈ 𝑆 ∀𝑓 ∈ (𝑎𝐽𝑏)∀𝑔 ∈ (𝑏𝐽𝑐)(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐶)𝑐)𝑓) ∈ (𝑎𝐽𝑐)))))
110	43, 108, 109	mpbir2and 726	1 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146   ≠ wne 2958  ∀wral 3079  Vcvv 3453   ⊆ wss 3899  ∅c0 4280  ifcif 4484  {csn 4587  ⟨cop 4837   class class class wbr 5105   × cxp 5652   Fn wfn 6541  ‘cfv 6546  (class class class)co 7427   ∈ cmpo 7429  Basecbs 17399  Hom chom 17451  compcco 17452  Catccat 17850  Idccid 17851  Hom_f chomf 17852   ⊆_cat cssc 17994  Subcatcsubc 17996

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-pm 8861  df-ixp 8937  df-cat 17854  df-cid 17855  df-homf 17856  df-ssc 17997  df-subc 17999

This theorem is used by:  iinfconstbaslem  50318