Theorem srhmsubc 20709

Description: According to df-subc 17828, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17856 and subcss2 17859). Therefore, the set of special ring homomorphisms (i.e., ring homomorphisms from a special ring to another ring of that kind) is a subcategory of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)

Hypotheses

Ref	Expression
srhmsubc.s	⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring
srhmsubc.c	⊢ 𝐶 = (𝑈 ∩ 𝑆)
srhmsubc.j	⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))

Assertion

Ref	Expression
srhmsubc	⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈)))

Distinct variable groups:   𝑆,𝑟   𝐶,𝑟,𝑠   𝑈,𝑟,𝑠   𝑉,𝑟,𝑠

Allowed substitution hints:   𝑆(𝑠)   𝐽(𝑠,𝑟)

Proof of Theorem srhmsubc

Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		srhmsubc.c	. . . 4 ⊢ 𝐶 = (𝑈 ∩ 𝑆)
2		eleq1w 2844	. . . . . . 7 ⊢ (𝑟 = 𝑥 → (𝑟 ∈ Ring ↔ 𝑥 ∈ Ring))
3		srhmsubc.s	. . . . . . 7 ⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring
4	2, 3	vtoclri 3549	. . . . . 6 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ Ring)
5	4	ssriv 3940	. . . . 5 ⊢ 𝑆 ⊆ Ring
6		sslin 4194	. . . . 5 ⊢ (𝑆 ⊆ Ring → (𝑈 ∩ 𝑆) ⊆ (𝑈 ∩ Ring))
7	5, 6	mp1i 13	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ 𝑆) ⊆ (𝑈 ∩ Ring))
8	1, 7	eqsstrid 3974	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ⊆ (𝑈 ∩ Ring))
9		ssid 3958	. . . . . 6 ⊢ (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦)
10		eqid 2761	. . . . . . 7 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈)
11		eqid 2761	. . . . . . 7 ⊢ (Base‘(RingCat‘𝑈)) = (Base‘(RingCat‘𝑈))
12		simpl 486	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑈 ∈ 𝑉)
13		eqid 2761	. . . . . . 7 ⊢ (Hom ‘(RingCat‘𝑈)) = (Hom ‘(RingCat‘𝑈))
14	3, 1	srhmsubclem2 20707	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
15	14	adantrr 727	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
16	3, 1	srhmsubclem2 20707	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
17	16	adantrl 726	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
18	10, 11, 12, 13, 15, 17	ringchom 20681	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(Hom ‘(RingCat‘𝑈))𝑦) = (𝑥 RingHom 𝑦))
19	9, 18	sseqtrrid 3979	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 RingHom 𝑦) ⊆ (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
20		srhmsubc.j	. . . . . . 7 ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))
21	20	a1i 11	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)))
22		oveq12 7401	. . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
23	22	adantl 485	. . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
24		simprl 780	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
25		simprr 782	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
26		ovexd 7427	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 RingHom 𝑦) ∈ V)
27	21, 23, 24, 25, 26	ovmpod 7544	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦))
28		eqid 2761	. . . . . 6 ⊢ (Homf ‘(RingCat‘𝑈)) = (Homf ‘(RingCat‘𝑈))
29	28, 11, 13, 15, 17	homfval 17707	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(Homf ‘(RingCat‘𝑈))𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
30	19, 27, 29	3sstr4d 3991	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))
31	30	ralrimivva 3204	. . 3 ⊢ (𝑈 ∈ 𝑉 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))
32		ovex 7425	. . . . . 6 ⊢ (𝑟 RingHom 𝑠) ∈ V
33	20, 32	fnmpoi 8047	. . . . 5 ⊢ 𝐽 Fn (𝐶 × 𝐶)
34	33	a1i 11	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐽 Fn (𝐶 × 𝐶))
35	28, 11	homffn 17708	. . . . 5 ⊢ (Homf ‘(RingCat‘𝑈)) Fn ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈)))
36		id 22	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ 𝑉)
37	10, 11, 36	ringcbas 20679	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → (Base‘(RingCat‘𝑈)) = (𝑈 ∩ Ring))
38	37	eqcomd 2767	. . . . . . 7 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) = (Base‘(RingCat‘𝑈)))
39	38	sqxpeqd 5677	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈))))
40	39	fneq2d 6611	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → ((Homf ‘(RingCat‘𝑈)) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) ↔ (Homf ‘(RingCat‘𝑈)) Fn ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈)))))
41	35, 40	mpbiri 260	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (Homf ‘(RingCat‘𝑈)) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
42		inex1g 5274	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V)
43	34, 41, 42	isssc 17836	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝐽 ⊆cat (Homf ‘(RingCat‘𝑈)) ↔ (𝐶 ⊆ (𝑈 ∩ Ring) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))))
44	8, 31, 43	mpbir2and 723	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ⊆cat (Homf ‘(RingCat‘𝑈)))
45	1	elin2 4155	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑆))
46	4	adantl 485	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ Ring)
47	45, 46	sylbi 219	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ Ring)
48	47	adantl 485	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ Ring)
49		eqid 2761	. . . . . . 7 ⊢ (Base‘𝑥) = (Base‘𝑥)
50	49	idrhm 20518	. . . . . 6 ⊢ (𝑥 ∈ Ring → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
51	48, 50	syl 17	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
52		eqid 2761	. . . . . 6 ⊢ (Id‘(RingCat‘𝑈)) = (Id‘(RingCat‘𝑈))
53		simpl 486	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑈 ∈ 𝑉)
54	10, 11, 52, 53, 14, 49	ringcid 20693	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → ((Id‘(RingCat‘𝑈))‘𝑥) = ( I ↾ (Base‘𝑥)))
55	20	a1i 11	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)))
56		oveq12 7401	. . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑥) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
57	56	adantl 485	. . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑥)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
58		simpr 488	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
59		ovexd 7427	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → (𝑥 RingHom 𝑥) ∈ V)
60	55, 57, 58, 58, 59	ovmpod 7544	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → (𝑥𝐽𝑥) = (𝑥 RingHom 𝑥))
61	51, 54, 60	3eltr4d 2876	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → ((Id‘(RingCat‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥))
62		eqid 2761	. . . . . . . . 9 ⊢ (comp‘(RingCat‘𝑈)) = (comp‘(RingCat‘𝑈))
63	10	ringccat 20692	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑉 → (RingCat‘𝑈) ∈ Cat)
64	63	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (RingCat‘𝑈) ∈ Cat)
65	14	adantr 484	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
66	65	adantr 484	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
67	16	ad2ant2r 757	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
68	67	adantr 484	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
69	3, 1	srhmsubclem2 20707	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
70	69	ad2ant2rl 759	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
71	70	adantr 484	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
72	53	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑈 ∈ 𝑉)
73		simpl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → 𝑦 ∈ 𝐶)
74	58, 73	anim12i 622	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶))
75	72, 74	jca 519	. . . . . . . . . . . . . 14 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)))
76	3, 1, 20	srhmsubclem3 20708	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐽𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
77	75, 76	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥𝐽𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
78	77	eleq2d 2847	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑓 ∈ (𝑥𝐽𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
79	78	biimpcd 251	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑥𝐽𝑦) → (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
80	79	adantr 484	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
81	80	impcom 411	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
82	3, 1, 20	srhmsubclem3 20708	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
83	82	adantlr 725	. . . . . . . . . . . . 13 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
84	83	eleq2d 2847	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) ↔ 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
85	84	biimpd 231	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
86	85	adantld 494	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
87	86	imp 410	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
88	11, 13, 62, 64, 66, 68, 71, 81, 87	catcocl 17700	. . . . . . . 8 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑧))
89	10, 11, 72, 13, 65, 70	ringchom 20681	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥(Hom ‘(RingCat‘𝑈))𝑧) = (𝑥 RingHom 𝑧))
90	89	eqcomd 2767	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥 RingHom 𝑧) = (𝑥(Hom ‘(RingCat‘𝑈))𝑧))
91	90	adantr 484	. . . . . . . 8 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑥 RingHom 𝑧) = (𝑥(Hom ‘(RingCat‘𝑈))𝑧))
92	88, 91	eleqtrrd 2864	. . . . . . 7 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥 RingHom 𝑧))
93	20	a1i 11	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)))
94		oveq12 7401	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑧) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
95	94	adantl 485	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑧)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
96	58	adantr 484	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
97		simprr 782	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ 𝐶)
98		ovexd 7427	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥 RingHom 𝑧) ∈ V)
99	93, 95, 96, 97, 98	ovmpod 7544	. . . . . . . 8 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
100	99	adantr 484	. . . . . . 7 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
101	92, 100	eleqtrrd 2864	. . . . . 6 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
102	101	ralrimivva 3204	. . . . 5 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
103	102	ralrimivva 3204	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
104	61, 103	jca 519	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → (((Id‘(RingCat‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
105	104	ralrimiva 3153	. 2 ⊢ (𝑈 ∈ 𝑉 → ∀𝑥 ∈ 𝐶 (((Id‘(RingCat‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
106	28, 52, 62, 63, 34	issubc2 17852	. 2 ⊢ (𝑈 ∈ 𝑉 → (𝐽 ∈ (Subcat‘(RingCat‘𝑈)) ↔ (𝐽 ⊆cat (Homf ‘(RingCat‘𝑈)) ∧ ∀𝑥 ∈ 𝐶 (((Id‘(RingCat‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
107	44, 105, 106	mpbir2and 723	1 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ∩ cin 3903   ⊆ wss 3904  ⟨cop 4587   class class class wbr 5099   I cid 5539   × cxp 5643   ↾ cres 5647   Fn wfn 6512  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  Basecbs 17228  Hom chom 17280  compcco 17281  Catccat 17679  Idccid 17680  Hom_f chomf 17681   ⊆_cat cssc 17823  Subcatcsubc 17825  Ringcrg 20262   RingHom crh 20497  RingCatcringc 20674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-map 8805  df-pm 8806  df-ixp 8876  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12479  df-z 12566  df-dec 12686  df-uz 12837  df-fz 13510  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17250  df-plusg 17282  df-hom 17293  df-cco 17294  df-0g 17453  df-cat 17683  df-cid 17684  df-homf 17685  df-ssc 17826  df-resc 17827  df-subc 17828  df-estrc 18138  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-mhm 18800  df-grp 18961  df-ghm 19237  df-mgp 20170  df-ur 20211  df-ring 20264  df-rhm 20500  df-ringc 20675

This theorem is referenced by:  sringcat  20710  crhmsubc  20711  drhmsubc  20810  fldhmsubc  20814