Theorem srhmsubc 20912

Description: According to df-subc 17967, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17995 and subcss2 17998). 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 3545	. . . . . 6 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ Ring)
5	4	ssriv 3935	. . . . 5 ⊢ 𝑆 ⊆ Ring
6		sslin 4188	. . . . 5 ⊢ (𝑆 ⊆ Ring → (𝑈 ∩ 𝑆) ⊆ (𝑈 ∩ Ring))
7	5, 6	mp1i 14	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ 𝑆) ⊆ (𝑈 ∩ Ring))
8	1, 7	eqsstrid 3969	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ⊆ (𝑈 ∩ Ring))
9		ssid 3953	. . . . . 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 488	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑈 ∈ 𝑉)
13		eqid 2761	. . . . . . 7 ⊢ (Hom ‘(RingCat‘𝑈)) = (Hom ‘(RingCat‘𝑈))
14	3, 1	srhmsubclem2 20910	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
15	14	adantrr 730	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
16	3, 1	srhmsubclem2 20910	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
17	16	adantrl 729	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
18	10, 11, 12, 13, 15, 17	ringchom 20884	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(Hom ‘(RingCat‘𝑈))𝑦) = (𝑥 RingHom 𝑦))
19	9, 18	sseqtrrid 3974	. . . . 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 7421	. . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
23	22	adantl 487	. . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
24		simprl 783	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
25		simprr 785	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
26		ovexd 7447	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 RingHom 𝑦) ∈ V)
27	21, 23, 24, 25, 26	ovmpod 7564	. . . . 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 17846	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(Homf ‘(RingCat‘𝑈))𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
30	19, 27, 29	3sstr4d 3986	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))
31	30	ralrimivva 3206	. . 3 ⊢ (𝑈 ∈ 𝑉 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))
32		ovex 7445	. . . . . 6 ⊢ (𝑟 RingHom 𝑠) ∈ V
33	20, 32	fnmpoi 8070	. . . . 5 ⊢ 𝐽 Fn (𝐶 × 𝐶)
34	33	a1i 11	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐽 Fn (𝐶 × 𝐶))
35	28, 11	homffn 17847	. . . . 5 ⊢ (Homf ‘(RingCat‘𝑈)) Fn ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈)))
36		id 23	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ 𝑉)
37	10, 11, 36	ringcbas 20882	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → (Base‘(RingCat‘𝑈)) = (𝑈 ∩ Ring))
38	37	eqcomd 2767	. . . . . . 7 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) = (Base‘(RingCat‘𝑈)))
39	38	sqxpeqd 5683	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈))))
40	39	fneq2d 6625	. . . . 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 261	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (Homf ‘(RingCat‘𝑈)) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
42		inex1g 5279	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V)
43	34, 41, 42	isssc 17975	. . 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 726	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ⊆cat (Homf ‘(RingCat‘𝑈)))
45	1	elin2 4149	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑆))
46	4	adantl 487	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ Ring)
47	45, 46	sylbi 220	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ Ring)
48	47	adantl 487	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ Ring)
49		eqid 2761	. . . . . . 7 ⊢ (Base‘𝑥) = (Base‘𝑥)
50	49	idrhm 20705	. . . . . 6 ⊢ (𝑥 ∈ Ring → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
51	48, 50	syl 18	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
52		eqid 2761	. . . . . 6 ⊢ (Id‘(RingCat‘𝑈)) = (Id‘(RingCat‘𝑈))
53		simpl 488	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑈 ∈ 𝑉)
54	10, 11, 52, 53, 14, 49	ringcid 20896	. . . . 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 7421	. . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑥) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
57	56	adantl 487	. . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑥)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
58		simpr 490	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
59		ovexd 7447	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → (𝑥 RingHom 𝑥) ∈ V)
60	55, 57, 58, 58, 59	ovmpod 7564	. . . . 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 20895	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑉 → (RingCat‘𝑈) ∈ Cat)
64	63	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (RingCat‘𝑈) ∈ Cat)
65	14	adantr 486	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
66	65	adantr 486	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
67	16	ad2ant2r 760	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
68	67	adantr 486	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
69	3, 1	srhmsubclem2 20910	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
70	69	ad2ant2rl 762	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
71	70	adantr 486	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
72	53	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑈 ∈ 𝑉)
73		simpl 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → 𝑦 ∈ 𝐶)
74	58, 73	anim12i 625	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶))
75	72, 74	jca 521	. . . . . . . . . . . . . 14 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)))
76	3, 1, 20	srhmsubclem3 20911	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐽𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
77	75, 76	syl 18	. . . . . . . . . . . . 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 252	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑥𝐽𝑦) → (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
80	79	adantr 486	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
81	80	impcom 413	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
82	3, 1, 20	srhmsubclem3 20911	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
83	82	adantlr 728	. . . . . . . . . . . . 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 232	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
86	85	adantld 496	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
87	86	imp 412	. . . . . . . . 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 17839	. . . . . . . 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 20884	. . . . . . . . . 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 486	. . . . . . . 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 7421	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑧) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
95	94	adantl 487	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑧)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
96	58	adantr 486	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
97		simprr 785	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ 𝐶)
98		ovexd 7447	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥 RingHom 𝑧) ∈ V)
99	93, 95, 96, 97, 98	ovmpod 7564	. . . . . . . 8 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
100	99	adantr 486	. . . . . . 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 3206	. . . . 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 3206	. . . 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 521	. . 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 3155	. 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 17991	. 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 726	1 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  ⟨cop 4590   class class class wbr 5103   I cid 5545   × cxp 5649   ↾ cres 5653   Fn wfn 6526  ‘cfv 6531  (class class class)co 7412   ∈ cmpo 7414  Basecbs 17367  Hom chom 17419  compcco 17420  Catccat 17818  Idccid 17819  Hom_f chomf 17820   ⊆_cat cssc 17962  Subcatcsubc 17964  Ringcrg 20439   RingHom crh 20679  RingCatcringc 20877

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-map 8833  df-pm 8834  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-4 12388  df-5 12389  df-6 12390  df-7 12391  df-8 12392  df-9 12393  df-n0 12588  df-z 12675  df-dec 12796  df-uz 12947  df-fz 13621  df-struct 17305  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-plusg 17421  df-hom 17432  df-cco 17433  df-0g 17592  df-cat 17822  df-cid 17823  df-homf 17824  df-ssc 17965  df-resc 17966  df-subc 17967  df-estrc 18277  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-mhm 18958  df-grp 19127  df-ghm 19408  df-mgp 20341  df-ur 20388  df-ring 20441  df-rhm 20682  df-ringc 20878

This theorem is used by:  sringcat  20913  crhmsubc  20914  drhmsubc  21018  fldhmsubc  21022