Theorem ssccatid 49338

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 49870). (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 2736	. . 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 17618	. . . . 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 17747	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐶))
10	1, 2, 3, 4, 9	rescbas 17755	. 2 ⊢ (𝜑 → 𝑆 = (Base‘𝐷))
11	1, 2, 3, 4, 9	reschom 17756	. 2 ⊢ (𝜑 → 𝐽 = (Hom ‘𝐷))
12		ssccatid.x	. . 3 ⊢ · = (comp‘𝐶)
13	1, 2, 3, 4, 9, 12	rescco 17758	. 2 ⊢ (𝜑 → · = (comp‘𝐷))
14	1	ovexi 7392	. . 3 ⊢ 𝐷 ∈ V
15	14	a1i 11	. 2 ⊢ (𝜑 → 𝐷 ∈ V)
16		biid 261	. 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 7366	. . . 4 ⊢ (𝑚 = 𝑓 → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑚) = ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓))
19		id 22	. . . 4 ⊢ (𝑚 = 𝑓 → 𝑚 = 𝑓)
20	18, 19	eqeq12d 2752	. . 3 ⊢ (𝑚 = 𝑓 → (( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑚) = 𝑚 ↔ ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓))
21		oveq1 7365	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝑎𝐽𝑏) = (𝑥𝐽𝑏))
22		opeq1 4829	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑏⟩)
23	22	oveq1d 7373	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (⟨𝑎, 𝑏⟩ · 𝑏) = (⟨𝑥, 𝑏⟩ · 𝑏))
24	23	oveqd 7375	. . . . . 6 ⊢ (𝑎 = 𝑥 → ( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = ( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚))
25	24	eqeq1d 2738	. . . . 5 ⊢ (𝑎 = 𝑥 → (( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = 𝑚 ↔ ( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚) = 𝑚))
26	21, 25	raleqbidv 3316	. . . 4 ⊢ (𝑎 = 𝑥 → (∀𝑚 ∈ (𝑎𝐽𝑏)( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = 𝑚 ↔ ∀𝑚 ∈ (𝑥𝐽𝑏)( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚) = 𝑚))
27		oveq2 7366	. . . . 5 ⊢ (𝑏 = 𝑦 → (𝑥𝐽𝑏) = (𝑥𝐽𝑦))
28		opeq2 4830	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ⟨𝑥, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
29		id 22	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → 𝑏 = 𝑦)
30	28, 29	oveq12d 7376	. . . . . . 7 ⊢ (𝑏 = 𝑦 → (⟨𝑥, 𝑏⟩ · 𝑏) = (⟨𝑥, 𝑦⟩ · 𝑦))
31	30	oveqd 7375	. . . . . 6 ⊢ (𝑏 = 𝑦 → ( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚) = ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑚))
32	31	eqeq1d 2738	. . . . 5 ⊢ (𝑏 = 𝑦 → (( 1 (⟨𝑥, 𝑏⟩ · 𝑏)𝑚) = 𝑚 ↔ ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑚) = 𝑚))
33	27, 32	raleqbidv 3316	. . . 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 3182	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 ∀𝑚 ∈ (𝑎𝐽𝑏)( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = 𝑚)
36	35	adantr 480	. . . 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 1231	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑥 ∈ 𝑆)
38		simpr1r 1232	. . . 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 1264	. . 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 7365	. . . 4 ⊢ (𝑚 = 𝑔 → (𝑚(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ))
43		id 22	. . . 4 ⊢ (𝑚 = 𝑔 → 𝑚 = 𝑔)
44	42, 43	eqeq12d 2752	. . 3 ⊢ (𝑚 = 𝑔 → ((𝑚(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑚 ↔ (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔))
45		oveq1 7365	. . . . 5 ⊢ (𝑎 = 𝑦 → (𝑎𝐽𝑏) = (𝑦𝐽𝑏))
46		id 22	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → 𝑎 = 𝑦)
47	46, 46	opeq12d 4837	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → ⟨𝑎, 𝑎⟩ = ⟨𝑦, 𝑦⟩)
48	47	oveq1d 7373	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (⟨𝑎, 𝑎⟩ · 𝑏) = (⟨𝑦, 𝑦⟩ · 𝑏))
49	48	oveqd 7375	. . . . . 6 ⊢ (𝑎 = 𝑦 → (𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = (𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ))
50	49	eqeq1d 2738	. . . . 5 ⊢ (𝑎 = 𝑦 → ((𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = 𝑚 ↔ (𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ) = 𝑚))
51	45, 50	raleqbidv 3316	. . . 4 ⊢ (𝑎 = 𝑦 → (∀𝑚 ∈ (𝑎𝐽𝑏)(𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = 𝑚 ↔ ∀𝑚 ∈ (𝑦𝐽𝑏)(𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ) = 𝑚))
52		oveq2 7366	. . . . 5 ⊢ (𝑏 = 𝑧 → (𝑦𝐽𝑏) = (𝑦𝐽𝑧))
53		oveq2 7366	. . . . . . 7 ⊢ (𝑏 = 𝑧 → (⟨𝑦, 𝑦⟩ · 𝑏) = (⟨𝑦, 𝑦⟩ · 𝑧))
54	53	oveqd 7375	. . . . . 6 ⊢ (𝑏 = 𝑧 → (𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ) = (𝑚(⟨𝑦, 𝑦⟩ · 𝑧) 1 ))
55	54	eqeq1d 2738	. . . . 5 ⊢ (𝑏 = 𝑧 → ((𝑚(⟨𝑦, 𝑦⟩ · 𝑏) 1 ) = 𝑚 ↔ (𝑚(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑚))
56	52, 55	raleqbidv 3316	. . . 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 3182	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 ∀𝑚 ∈ (𝑎𝐽𝑏)(𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = 𝑚)
59	58	adantr 480	. . . 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 1233	. . . 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 1265	. . 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 482	. . 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 1390	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))
67		eqid 2736	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
68	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝐶 ∈ Cat)
69	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑆 ⊆ (Base‘𝐶))
70	69, 37	sseldd 3934	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑥 ∈ (Base‘𝐶))
71	69, 38	sseldd 3934	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑦 ∈ (Base‘𝐶))
72	69, 60	sseldd 3934	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑧 ∈ (Base‘𝐶))
73	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝐽 Fn (𝑆 × 𝑆))
74	8	adantr 480	. . . . . 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 17748	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑥𝐽𝑦) ⊆ (𝑥𝐻𝑦))
76	75, 40	sseldd 3934	. . . 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 17617	. . . 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 2838	. . 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 17748	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑦𝐽𝑧) ⊆ (𝑦𝐻𝑧))
80	79, 62	sseldd 3934	. . . 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 17617	. . . 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 2838	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
83		simpr2r 1234	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑤 ∈ 𝑆)
84	69, 83	sseldd 3934	. . 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 17748	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → (𝑧𝐽𝑤) ⊆ (𝑧𝐻𝑤))
86		simpr33 1266	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧) ∧ 𝑘 ∈ (𝑧𝐽𝑤)))) → 𝑘 ∈ (𝑧𝐽𝑤))
87	85, 86	sseldd 3934	. . . 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 17617	. . . 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 2838	. . 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 17611	. 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 17606	1 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ (Id‘𝐷) = (𝑦 ∈ 𝑆 ↦ 1 )))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ⊆ wss 3901  ⟨cop 4586   class class class wbr 5098   ↦ cmpt 5179   × cxp 5622   Fn wfn 6487  ‘cfv 6492  (class class class)co 7358  Basecbs 17138  Hom chom 17190  compcco 17191  Catccat 17589  Idccid 17590  Hom_f chomf 17591   ⊆_cat cssc 17733   ↾_cat cresc 17734

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-ixp 8838  df-en 8886  df-dom 8887  df-sdom 8888  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12610  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-hom 17203  df-cco 17204  df-cat 17593  df-cid 17594  df-homf 17595  df-ssc 17736  df-resc 17737

This theorem is referenced by: (None)