Theorem ssccatid 50389

Description: A category 𝐶 restricted by 𝐽 is a category if all of the following are satisfied: a) the base is a subset of base of 𝐶, b) all hom-sets are subsets of hom-sets of 𝐶, c) it has identity morphisms for all objects, d) the composition under 𝐶 is closed in 𝐽. But 𝐽 might not be a subcategory of 𝐶 (see cnelsubc 50945). (Contributed by Zhi Wang, 6-Nov-2025.)

Hypotheses

Ref	Expression
ssccatid.h	⊢ 𝐻 = (Homf ‘𝐶)
ssccatid.d	⊢ 𝐷 = (𝐶 ↾cat 𝐽)
ssccatid.x	⊢ · = (comp‘𝐶)
ssccatid.j	⊢ (𝜑 → 𝐽 ⊆cat 𝐻)
ssccatid.f	⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
ssccatid.c	⊢ (𝜑 → 𝐶 ∈ Cat)
ssccatid.i	⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 1 ∈ (𝑦𝐽𝑦))
ssccatid.l	⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ∧ 𝑚 ∈ (𝑎𝐽𝑏))) → ( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = 𝑚)
ssccatid.r	⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ∧ 𝑚 ∈ (𝑎𝐽𝑏))) → (𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = 𝑚)
ssccatid.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))

Assertion

Ref	Expression
ssccatid	⊢ (𝜑 → (𝐷 ∈ Cat ∧ (Id‘𝐷) = (𝑦 ∈ 𝑆 ↦ 1 )))

Distinct variable groups:   1 ,𝑎,𝑏,𝑚,𝑥   1 ,𝑓,𝑔,𝑧,𝑥   · ,𝑎,𝑏,𝑚,𝑥,𝑦   · ,𝑓,𝑔,𝑧,𝑦   𝐷,𝑔,𝑦,𝑧   𝐽,𝑎,𝑏,𝑚,𝑥,𝑦   𝑓,𝐽,𝑔,𝑧   𝑆,𝑎,𝑏,𝑚,𝑥,𝑦   𝑆,𝑓,𝑔,𝑧   𝜑,𝑎,𝑏,𝑚,𝑥,𝑦   𝑧,𝑏,𝑚,𝜑   𝑓,𝑚,𝜑,𝑔

Allowed substitution hints:   𝐶(𝑥, 𝑦, 𝑧, 𝑓, 𝑔, 𝑚, 𝑎, 𝑏)   𝐷(𝑥, 𝑓, 𝑚, 𝑎, 𝑏)   1 (𝑦)   𝐻(𝑥, 𝑦, 𝑧, 𝑓, 𝑔, 𝑚, 𝑎, 𝑏)

Proof of Theorem ssccatid

Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssccatid.d	. . 3 ⊢ 𝐷 = (𝐶 ↾cat 𝐽)
2		eqid 2763	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		ssccatid.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		ssccatid.f	. . 3 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
5		ssccatid.h	. . . . . 6 ⊢ 𝐻 = (Homf ‘𝐶)
6	5, 2	homffn 17893	. . . . 5 ⊢ 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶))
7	6	a1i 11	. . . 4 ⊢ (𝜑 → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)))
8		ssccatid.j	. . . 4 ⊢ (𝜑 → 𝐽 ⊆cat 𝐻)
9	4, 7, 8	ssc1 18022	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐶))
10	1, 2, 3, 4, 9	rescbas 18030	. 2 ⊢ (𝜑 → 𝑆 = (Base‘𝐷))
11	1, 2, 3, 4, 9	reschom 18031	. 2 ⊢ (𝜑 → 𝐽 = (Hom ‘𝐷))
12		ssccatid.x	. . 3 ⊢ · = (comp‘𝐶)
13	1, 2, 3, 4, 9, 12	rescco 18033	. 2 ⊢ (𝜑 → · = (comp‘𝐷))
14	1	ovexi 7474	. . 3 ⊢ 𝐷 ∈ V
15	14	a1i 11	. 2 ⊢ (𝜑 → 𝐷 ∈ V)
16		biid 264	. 2 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤))) ↔ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤))))
17		ssccatid.i	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 1 ∈ (𝑦𝐽𝑦))
18		oveq2 7448	. . . 4 ⊢ (𝑚 = 𝑓 → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑚) = ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓))
19		id 23	. . . 4 ⊢ (𝑚 = 𝑓 → 𝑚 = 𝑓)
20	18, 19	eqeq12d 2779	. . 3 ⊢ (𝑚 = 𝑓 → (( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑚) = 𝑚 ↔ ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓))
21		oveq1 7447	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝑎𝐽𝑏) = (𝑥𝐽𝑏))
22		opeq1 4843	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑏⟩)
23	22	oveq1d 7455	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (⟨𝑎, 𝑏⟩ · 𝑏) = (⟨𝑥, 𝑏⟩ · 𝑏))
24	23	oveqd 7457	. . . . . 6 ⊢ (𝑎 = 𝑥 → ( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = ( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚))
25	24	eqeq1d 2765	. . . . 5 ⊢ (𝑎 = 𝑥 → (( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = 𝑚 ↔ ( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚) = 𝑚))
26	21, 25	raleqbidv 3338	. . . 4 ⊢ (𝑎 = 𝑥 → (∀𝑚 ∈ (𝑎𝐽𝑏)( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = 𝑚 ↔ ∀𝑚 ∈ (𝑥𝐽𝑏)( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚) = 𝑚))
27		oveq2 7448	. . . . 5 ⊢ (𝑏 = 𝑦 → (𝑥𝐽𝑏) = (𝑥𝐽𝑦))
28		opeq2 4844	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ⟨𝑥, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
29		id 23	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → 𝑏 = 𝑦)
30	28, 29	oveq12d 7458	. . . . . . 7 ⊢ (𝑏 = 𝑦 → (⟨𝑥, 𝑏⟩ · 𝑏) = (⟨𝑥, 𝑦⟩ · 𝑦))
31	30	oveqd 7457	. . . . . 6 ⊢ (𝑏 = 𝑦 → ( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚) = ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑚))
32	31	eqeq1d 2765	. . . . 5 ⊢ (𝑏 = 𝑦 → (( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚) = 𝑚 ↔ ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑚) = 𝑚))
33	27, 32	raleqbidv 3338	. . . 4 ⊢ (𝑏 = 𝑦 → (∀𝑚 ∈ (𝑥𝐽𝑏)( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚) = 𝑚 ↔ ∀𝑚 ∈ (𝑥𝐽𝑦)( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑚) = 𝑚))
34		ssccatid.l	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ∧ 𝑚 ∈ (𝑎𝐽𝑏))) → ( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = 𝑚)
35	34	ralrimivvva 3212	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 ∀𝑚 ∈ (𝑎𝐽𝑏)( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = 𝑚)
36	35	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 ∀𝑚 ∈ (𝑎𝐽𝑏)( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = 𝑚)
37		simpr1l 1249	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑥 ∈ 𝑆)
38		simpr1r 1250	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑦 ∈ 𝑆)
39	26, 33, 36, 37, 38	rspc2dv 3594	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → ∀𝑚 ∈ (𝑥𝐽𝑦)( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑚) = 𝑚)
40		simpr31 1282	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑓 ∈ (𝑥𝐽𝑦))
41	20, 39, 40	rspcdva 3581	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓)
42		oveq1 7447	. . . 4 ⊢ (𝑚 = 𝑔 → (𝑚(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ))
43		id 23	. . . 4 ⊢ (𝑚 = 𝑔 → 𝑚 = 𝑔)
44	42, 43	eqeq12d 2779	. . 3 ⊢ (𝑚 = 𝑔 → ((𝑚(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑚 ↔ (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔))
45		oveq1 7447	. . . . 5 ⊢ (𝑎 = 𝑦 → (𝑎𝐽𝑏) = (𝑦𝐽𝑏))
46		id 23	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → 𝑎 = 𝑦)
47	46, 46	opeq12d 4851	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → ⟨𝑎, 𝑎⟩ = ⟨𝑦, 𝑦⟩)
48	47	oveq1d 7455	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (⟨𝑎, 𝑎⟩ · 𝑏) = (⟨𝑦, 𝑦⟩ · 𝑏))
49	48	oveqd 7457	. . . . . 6 ⊢ (𝑎 = 𝑦 → (𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = (𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ))
50	49	eqeq1d 2765	. . . . 5 ⊢ (𝑎 = 𝑦 → ((𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = 𝑚 ↔ (𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ) = 𝑚))
51	45, 50	raleqbidv 3338	. . . 4 ⊢ (𝑎 = 𝑦 → (∀𝑚 ∈ (𝑎𝐽𝑏)(𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = 𝑚 ↔ ∀𝑚 ∈ (𝑦𝐽𝑏)(𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ) = 𝑚))
52		oveq2 7448	. . . . 5 ⊢ (𝑏 = 𝑧 → (𝑦𝐽𝑏) = (𝑦𝐽𝑧))
53		oveq2 7448	. . . . . . 7 ⊢ (𝑏 = 𝑧 → (⟨𝑦, 𝑦⟩ · 𝑏) = (⟨𝑦, 𝑦⟩ · 𝑧))
54	53	oveqd 7457	. . . . . 6 ⊢ (𝑏 = 𝑧 → (𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ) = (𝑚(⟨𝑦, 𝑦⟩ · 𝑧) 1 ))
55	54	eqeq1d 2765	. . . . 5 ⊢ (𝑏 = 𝑧 → ((𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ) = 𝑚 ↔ (𝑚(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑚))
56	52, 55	raleqbidv 3338	. . . 4 ⊢ (𝑏 = 𝑧 → (∀𝑚 ∈ (𝑦𝐽𝑏)(𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ) = 𝑚 ↔ ∀𝑚 ∈ (𝑦𝐽𝑧)(𝑚(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑚))
57		ssccatid.r	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ∧ 𝑚 ∈ (𝑎𝐽𝑏))) → (𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = 𝑚)
58	57	ralrimivvva 3212	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 ∀𝑚 ∈ (𝑎𝐽𝑏)(𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = 𝑚)
59	58	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 ∀𝑚 ∈ (𝑎𝐽𝑏)(𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = 𝑚)
60		simpr2l 1251	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑧 ∈ 𝑆)
61	51, 56, 59, 38, 60	rspc2dv 3594	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → ∀𝑚 ∈ (𝑦𝐽𝑧)(𝑚(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑚)
62		simpr32 1283	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑔 ∈ (𝑦𝐽𝑧))
63	44, 61, 62	rspcdva 3581	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔)
64		simpl 488	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝜑)
65		ssccatid.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))
66	64, 37, 38, 60, 40, 62, 65	syl132anc 1416	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))
67		eqid 2763	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
68	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝐶 ∈ Cat)
69	9	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑆 ⊆ (Base‘𝐶))
70	69, 37	sseldd 3914	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑥 ∈ (Base‘𝐶))
71	69, 38	sseldd 3914	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑦 ∈ (Base‘𝐶))
72	69, 60	sseldd 3914	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑧 ∈ (Base‘𝐶))
73	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝐽 Fn (𝑆 × 𝑆))
74	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝐽 ⊆cat 𝐻)
75	73, 74, 37, 38	ssc2 18023	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑥𝐽𝑦) ⊆ (𝑥𝐻𝑦))
76	75, 40	sseldd 3914	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑓 ∈ (𝑥𝐻𝑦))
77	5, 2, 67, 70, 71	homfval 17892	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
78	76, 77	eleqtrd 2866	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
79	73, 74, 38, 60	ssc2 18023	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑦𝐽𝑧) ⊆ (𝑦𝐻𝑧))
80	79, 62	sseldd 3914	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑔 ∈ (𝑦𝐻𝑧))
81	5, 2, 67, 71, 72	homfval 17892	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑦𝐻𝑧) = (𝑦(Hom ‘𝐶)𝑧))
82	80, 81	eleqtrd 2866	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
83		simpr2r 1252	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑤 ∈ 𝑆)
84	69, 83	sseldd 3914	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑤 ∈ (Base‘𝐶))
85	73, 74, 60, 83	ssc2 18023	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑧𝐽𝑤) ⊆ (𝑧𝐻𝑤))
86		simpr33 1284	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑘 ∈ (𝑧𝐽𝑤))
87	85, 86	sseldd 3914	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑘 ∈ (𝑧𝐻𝑤))
88	5, 2, 67, 72, 84	homfval 17892	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑧𝐻𝑤) = (𝑧(Hom ‘𝐶)𝑤))
89	87, 88	eleqtrd 2866	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤))
90	2, 67, 12, 68, 70, 71, 72, 78, 82, 84, 89	catass 17886	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))
91	10, 11, 13, 15, 16, 17, 41, 63, 66, 90	iscatd2 17881	1 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ (Id‘𝐷) = (𝑦 ∈ 𝑆 ↦ 1 )))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  ⟨cop 4838   class class class wbr 5106   ↦ cmpt 5189   × cxp 5653   Fn wfn 6550  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  Hom chom 17465  compcco 17466  Catccat 17864  Idccid 17865  Hom_f chomf 17866   ⊆_cat cssc 18008   ↾_cat cresc 18009

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-er 8742  df-ixp 8951  df-en 8999  df-dom 9000  df-sdom 9001  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-hom 17478  df-cco 17479  df-cat 17868  df-cid 17869  df-homf 17870  df-ssc 18011  df-resc 18012

This theorem is used by: (None)