Theorem srhmsubc 20766

Description: According to df-subc 17870, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17898 and subcss2 17901). 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 2846	. . . . . . 7 ⊢ (𝑟 = 𝑥 → (𝑟 ∈ Ring ↔ 𝑥 ∈ Ring))
3		srhmsubc.s	. . . . . . 7 ⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring
4	2, 3	vtoclri 3550	. . . . . 6 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ Ring)
5	4	ssriv 3942	. . . . 5 ⊢ 𝑆 ⊆ Ring
6		sslin 4196	. . . . 5 ⊢ (𝑆 ⊆ Ring → (𝑈 ∩ 𝑆) ⊆ (𝑈 ∩ Ring))
7	5, 6	mp1i 14	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ 𝑆) ⊆ (𝑈 ∩ Ring))
8	1, 7	eqsstrid 3976	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ⊆ (𝑈 ∩ Ring))
9		ssid 3960	. . . . . 6 ⊢ (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦)
10		eqid 2763	. . . . . . 7 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈)
11		eqid 2763	. . . . . . 7 ⊢ (Base‘(RingCat‘𝑈)) = (Base‘(RingCat‘𝑈))
12		simpl 487	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑈 ∈ 𝑉)
13		eqid 2763	. . . . . . 7 ⊢ (Hom ‘(RingCat‘𝑈)) = (Hom ‘(RingCat‘𝑈))
14	3, 1	srhmsubclem2 20764	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
15	14	adantrr 729	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
16	3, 1	srhmsubclem2 20764	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
17	16	adantrl 728	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
18	10, 11, 12, 13, 15, 17	ringchom 20738	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(Hom ‘(RingCat‘𝑈))𝑦) = (𝑥 RingHom 𝑦))
19	9, 18	sseqtrrid 3981	. . . . 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 486	. . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
24		simprl 782	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
25		simprr 784	. . . . . 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 2763	. . . . . 6 ⊢ (Homf ‘(RingCat‘𝑈)) = (Homf ‘(RingCat‘𝑈))
29	28, 11, 13, 15, 17	homfval 17749	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(Homf ‘(RingCat‘𝑈))𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
30	19, 27, 29	3sstr4d 3993	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))
31	30	ralrimivva 3208	. . 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 8068	. . . . 5 ⊢ 𝐽 Fn (𝐶 × 𝐶)
34	33	a1i 11	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐽 Fn (𝐶 × 𝐶))
35	28, 11	homffn 17750	. . . . 5 ⊢ (Homf ‘(RingCat‘𝑈)) Fn ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈)))
36		id 23	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ 𝑉)
37	10, 11, 36	ringcbas 20736	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → (Base‘(RingCat‘𝑈)) = (𝑈 ∩ Ring))
38	37	eqcomd 2769	. . . . . . 7 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) = (Base‘(RingCat‘𝑈)))
39	38	sqxpeqd 5695	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈))))
40	39	fneq2d 6631	. . . . 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 5289	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V)
43	34, 41, 42	isssc 17878	. . 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 725	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ⊆cat (Homf ‘(RingCat‘𝑈)))
45	1	elin2 4157	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑆))
46	4	adantl 486	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ Ring)
47	45, 46	sylbi 220	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ Ring)
48	47	adantl 486	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ Ring)
49		eqid 2763	. . . . . . 7 ⊢ (Base‘𝑥) = (Base‘𝑥)
50	49	idrhm 20573	. . . . . 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 2763	. . . . . 6 ⊢ (Id‘(RingCat‘𝑈)) = (Id‘(RingCat‘𝑈))
53		simpl 487	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑈 ∈ 𝑉)
54	10, 11, 52, 53, 14, 49	ringcid 20750	. . . . 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 486	. . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑥)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
58		simpr 489	. . . . . 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 2878	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → ((Id‘(RingCat‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥))
62		eqid 2763	. . . . . . . . 9 ⊢ (comp‘(RingCat‘𝑈)) = (comp‘(RingCat‘𝑈))
63	10	ringccat 20749	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑉 → (RingCat‘𝑈) ∈ Cat)
64	63	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (RingCat‘𝑈) ∈ Cat)
65	14	adantr 485	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
66	65	adantr 485	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
67	16	ad2ant2r 759	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
68	67	adantr 485	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
69	3, 1	srhmsubclem2 20764	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
70	69	ad2ant2rl 761	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
71	70	adantr 485	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
72	53	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑈 ∈ 𝑉)
73		simpl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → 𝑦 ∈ 𝐶)
74	58, 73	anim12i 624	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶))
75	72, 74	jca 520	. . . . . . . . . . . . . 14 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)))
76	3, 1, 20	srhmsubclem3 20765	. . . . . . . . . . . . . 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 2849	. . . . . . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
81	80	impcom 412	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
82	3, 1, 20	srhmsubclem3 20765	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
83	82	adantlr 727	. . . . . . . . . . . . 13 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
84	83	eleq2d 2849	. . . . . . . . . . . 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 495	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
87	86	imp 411	. . . . . . . . 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 17742	. . . . . . . 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 20738	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥(Hom ‘(RingCat‘𝑈))𝑧) = (𝑥 RingHom 𝑧))
90	89	eqcomd 2769	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥 RingHom 𝑧) = (𝑥(Hom ‘(RingCat‘𝑈))𝑧))
91	90	adantr 485	. . . . . . . 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 2866	. . . . . . 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 486	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑧)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
96	58	adantr 485	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
97		simprr 784	. . . . . . . . 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 485	. . . . . . 7 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
101	92, 100	eleqtrrd 2866	. . . . . 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 3208	. . . . 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 3208	. . . 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 520	. . 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 3157	. 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 17894	. 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 725	1 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∩ cin 3905   ⊆ wss 3906  ⟨cop 4596   class class class wbr 5110   I cid 5557   × cxp 5661   ↾ cres 5665   Fn wfn 6533  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414  Basecbs 17270  Hom chom 17322  compcco 17323  Catccat 17721  Idccid 17722  Hom_f chomf 17723   ⊆_cat cssc 17865  Subcatcsubc 17867  Ringcrg 20316   RingHom crh 20552  RingCatcringc 20731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-map 8827  df-pm 8828  df-ixp 8897  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-z 12593  df-dec 12713  df-uz 12864  df-fz 13537  df-struct 17208  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-hom 17335  df-cco 17336  df-0g 17495  df-cat 17725  df-cid 17726  df-homf 17727  df-ssc 17868  df-resc 17869  df-subc 17870  df-estrc 18180  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-mhm 18842  df-grp 19004  df-ghm 19285  df-mgp 20218  df-ur 20265  df-ring 20318  df-rhm 20555  df-ringc 20732

This theorem is referenced by:  sringcat  20767  crhmsubc  20768  drhmsubc  20865  fldhmsubc  20869