Theorem ssccatid 49998

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 50530). (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 2760	. . 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 17781	. . . . 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 17910	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐶))
10	1, 2, 3, 4, 9	rescbas 17918	. 2 ⊢ (𝜑 → 𝑆 = (Base‘𝐷))
11	1, 2, 3, 4, 9	reschom 17919	. 2 ⊢ (𝜑 → 𝐽 = (Hom ‘𝐷))
12		ssccatid.x	. . 3 ⊢ · = (comp‘𝐶)
13	1, 2, 3, 4, 9, 12	rescco 17921	. 2 ⊢ (𝜑 → · = (comp‘𝐷))
14	1	ovexi 7447	. . 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 7421	. . . 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 2776	. . 3 ⊢ (𝑚 = 𝑓 → (( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑚) = 𝑚 ↔ ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓))
21		oveq1 7420	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝑎𝐽𝑏) = (𝑥𝐽𝑏))
22		opeq1 4833	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑏⟩)
23	22	oveq1d 7428	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (⟨𝑎, 𝑏⟩ · 𝑏) = (⟨𝑥, 𝑏⟩ · 𝑏))
24	23	oveqd 7430	. . . . . 6 ⊢ (𝑎 = 𝑥 → ( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = ( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚))
25	24	eqeq1d 2762	. . . . 5 ⊢ (𝑎 = 𝑥 → (( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = 𝑚 ↔ ( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚) = 𝑚))
26	21, 25	raleqbidv 3334	. . . 4 ⊢ (𝑎 = 𝑥 → (∀𝑚 ∈ (𝑎𝐽𝑏)( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = 𝑚 ↔ ∀𝑚 ∈ (𝑥𝐽𝑏)( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚) = 𝑚))
27		oveq2 7421	. . . . 5 ⊢ (𝑏 = 𝑦 → (𝑥𝐽𝑏) = (𝑥𝐽𝑦))
28		opeq2 4834	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ⟨𝑥, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
29		id 23	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → 𝑏 = 𝑦)
30	28, 29	oveq12d 7431	. . . . . . 7 ⊢ (𝑏 = 𝑦 → (⟨𝑥, 𝑏⟩ · 𝑏) = (⟨𝑥, 𝑦⟩ · 𝑦))
31	30	oveqd 7430	. . . . . 6 ⊢ (𝑏 = 𝑦 → ( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚) = ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑚))
32	31	eqeq1d 2762	. . . . 5 ⊢ (𝑏 = 𝑦 → (( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚) = 𝑚 ↔ ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑚) = 𝑚))
33	27, 32	raleqbidv 3334	. . . 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 3208	. . . . 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 3591	. . 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 3577	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓)
42		oveq1 7420	. . . 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 2776	. . 3 ⊢ (𝑚 = 𝑔 → ((𝑚(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑚 ↔ (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔))
45		oveq1 7420	. . . . 5 ⊢ (𝑎 = 𝑦 → (𝑎𝐽𝑏) = (𝑦𝐽𝑏))
46		id 23	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → 𝑎 = 𝑦)
47	46, 46	opeq12d 4841	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → ⟨𝑎, 𝑎⟩ = ⟨𝑦, 𝑦⟩)
48	47	oveq1d 7428	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (⟨𝑎, 𝑎⟩ · 𝑏) = (⟨𝑦, 𝑦⟩ · 𝑏))
49	48	oveqd 7430	. . . . . 6 ⊢ (𝑎 = 𝑦 → (𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = (𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ))
50	49	eqeq1d 2762	. . . . 5 ⊢ (𝑎 = 𝑦 → ((𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = 𝑚 ↔ (𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ) = 𝑚))
51	45, 50	raleqbidv 3334	. . . 4 ⊢ (𝑎 = 𝑦 → (∀𝑚 ∈ (𝑎𝐽𝑏)(𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = 𝑚 ↔ ∀𝑚 ∈ (𝑦𝐽𝑏)(𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ) = 𝑚))
52		oveq2 7421	. . . . 5 ⊢ (𝑏 = 𝑧 → (𝑦𝐽𝑏) = (𝑦𝐽𝑧))
53		oveq2 7421	. . . . . . 7 ⊢ (𝑏 = 𝑧 → (⟨𝑦, 𝑦⟩ · 𝑏) = (⟨𝑦, 𝑦⟩ · 𝑧))
54	53	oveqd 7430	. . . . . 6 ⊢ (𝑏 = 𝑧 → (𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ) = (𝑚(⟨𝑦, 𝑦⟩ · 𝑧) 1 ))
55	54	eqeq1d 2762	. . . . 5 ⊢ (𝑏 = 𝑧 → ((𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ) = 𝑚 ↔ (𝑚(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑚))
56	52, 55	raleqbidv 3334	. . . 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 3208	. . . . 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 3591	. . 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 3577	. 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 1415	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))
67		eqid 2760	. . 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 3932	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑥 ∈ (Base‘𝐶))
71	69, 38	sseldd 3932	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑦 ∈ (Base‘𝐶))
72	69, 60	sseldd 3932	. . 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 17911	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑥𝐽𝑦) ⊆ (𝑥𝐻𝑦))
76	75, 40	sseldd 3932	. . . 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 17780	. . . 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 2862	. . 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 17911	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑦𝐽𝑧) ⊆ (𝑦𝐻𝑧))
80	79, 62	sseldd 3932	. . . 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 17780	. . . 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 2862	. . 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 3932	. . 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 17911	. . . . 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 3932	. . . 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 17780	. . . 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 2862	. . 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 17774	. 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 17769	1 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ (Id‘𝐷) = (𝑦 ∈ 𝑆 ↦ 1 )))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ⊆ wss 3899  ⟨cop 4590   class class class wbr 5103   ↦ cmpt 5186   × cxp 5653   Fn wfn 6528  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  Hom chom 17353  compcco 17354  Catccat 17752  Idccid 17753  Hom_f chomf 17754   ⊆_cat cssc 17896   ↾_cat cresc 17897

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 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  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 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 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-z 12616  df-dec 12737  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-hom 17366  df-cco 17367  df-cat 17756  df-cid 17757  df-homf 17758  df-ssc 17899  df-resc 17900

This theorem is used by: (None)