Theorem srhmsubc 45634

Description: According to df-subc 17524, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17555 and subcss2 17558). 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 2821	. . . . . . 7 ⊢ (𝑟 = 𝑥 → (𝑟 ∈ Ring ↔ 𝑥 ∈ Ring))
3		srhmsubc.s	. . . . . . 7 ⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring
4	2, 3	vtoclri 3525	. . . . . 6 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ Ring)
5	4	ssriv 3925	. . . . 5 ⊢ 𝑆 ⊆ Ring
6		sslin 4168	. . . . 5 ⊢ (𝑆 ⊆ Ring → (𝑈 ∩ 𝑆) ⊆ (𝑈 ∩ Ring))
7	5, 6	mp1i 13	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ 𝑆) ⊆ (𝑈 ∩ Ring))
8	1, 7	eqsstrid 3969	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ⊆ (𝑈 ∩ Ring))
9		ssid 3943	. . . . . 6 ⊢ (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦)
10		eqid 2738	. . . . . . 7 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈)
11		eqid 2738	. . . . . . 7 ⊢ (Base‘(RingCat‘𝑈)) = (Base‘(RingCat‘𝑈))
12		simpl 483	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑈 ∈ 𝑉)
13		eqid 2738	. . . . . . 7 ⊢ (Hom ‘(RingCat‘𝑈)) = (Hom ‘(RingCat‘𝑈))
14	3, 1	srhmsubclem2 45632	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
15	14	adantrr 714	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
16	3, 1	srhmsubclem2 45632	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
17	16	adantrl 713	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
18	10, 11, 12, 13, 15, 17	ringchom 45571	. . . . . 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 7284	. . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
23	22	adantl 482	. . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
24		simprl 768	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
25		simprr 770	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
26		ovexd 7310	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 RingHom 𝑦) ∈ V)
27	21, 23, 24, 25, 26	ovmpod 7425	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦))
28		eqid 2738	. . . . . 6 ⊢ (Homf ‘(RingCat‘𝑈)) = (Homf ‘(RingCat‘𝑈))
29	28, 11, 13, 15, 17	homfval 17401	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(Homf ‘(RingCat‘𝑈))𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
30	19, 27, 29	3sstr4d 3968	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))
31	30	ralrimivva 3123	. . 3 ⊢ (𝑈 ∈ 𝑉 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))
32		ovex 7308	. . . . . 6 ⊢ (𝑟 RingHom 𝑠) ∈ V
33	20, 32	fnmpoi 7910	. . . . 5 ⊢ 𝐽 Fn (𝐶 × 𝐶)
34	33	a1i 11	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐽 Fn (𝐶 × 𝐶))
35	28, 11	homffn 17402	. . . . 5 ⊢ (Homf ‘(RingCat‘𝑈)) Fn ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈)))
36		id 22	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ 𝑉)
37	10, 11, 36	ringcbas 45569	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → (Base‘(RingCat‘𝑈)) = (𝑈 ∩ Ring))
38	37	eqcomd 2744	. . . . . . 7 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) = (Base‘(RingCat‘𝑈)))
39	38	sqxpeqd 5621	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈))))
40	39	fneq2d 6527	. . . . 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 257	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (Homf ‘(RingCat‘𝑈)) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
42		inex1g 5243	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V)
43	34, 41, 42	isssc 17532	. . 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 710	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ⊆cat (Homf ‘(RingCat‘𝑈)))
45	1	elin2 4131	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑆))
46	4	adantl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ Ring)
47	45, 46	sylbi 216	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ Ring)
48	47	adantl 482	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ Ring)
49		eqid 2738	. . . . . . 7 ⊢ (Base‘𝑥) = (Base‘𝑥)
50	49	idrhm 19975	. . . . . 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 2738	. . . . . 6 ⊢ (Id‘(RingCat‘𝑈)) = (Id‘(RingCat‘𝑈))
53		simpl 483	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑈 ∈ 𝑉)
54	10, 11, 52, 53, 14, 49	ringcid 45583	. . . . 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 7284	. . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑥) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
57	56	adantl 482	. . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑥)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
58		simpr 485	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
59		ovexd 7310	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → (𝑥 RingHom 𝑥) ∈ V)
60	55, 57, 58, 58, 59	ovmpod 7425	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → (𝑥𝐽𝑥) = (𝑥 RingHom 𝑥))
61	51, 54, 60	3eltr4d 2854	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → ((Id‘(RingCat‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥))
62		eqid 2738	. . . . . . . . 9 ⊢ (comp‘(RingCat‘𝑈)) = (comp‘(RingCat‘𝑈))
63	10	ringccat 45582	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑉 → (RingCat‘𝑈) ∈ Cat)
64	63	ad3antrrr 727	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (RingCat‘𝑈) ∈ Cat)
65	14	adantr 481	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
66	65	adantr 481	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
67	16	ad2ant2r 744	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
68	67	adantr 481	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
69	3, 1	srhmsubclem2 45632	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
70	69	ad2ant2rl 746	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
71	70	adantr 481	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
72	53	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑈 ∈ 𝑉)
73		simpl 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → 𝑦 ∈ 𝐶)
74	58, 73	anim12i 613	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶))
75	72, 74	jca 512	. . . . . . . . . . . . . 14 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)))
76	3, 1, 20	srhmsubclem3 45633	. . . . . . . . . . . . . 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 2824	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑓 ∈ (𝑥𝐽𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
79	78	biimpcd 248	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑥𝐽𝑦) → (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
80	79	adantr 481	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
81	80	impcom 408	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
82	3, 1, 20	srhmsubclem3 45633	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
83	82	adantlr 712	. . . . . . . . . . . . 13 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
84	83	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) ↔ 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
85	84	biimpd 228	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
86	85	adantld 491	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
87	86	imp 407	. . . . . . . . 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 17394	. . . . . . . 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 45571	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥(Hom ‘(RingCat‘𝑈))𝑧) = (𝑥 RingHom 𝑧))
90	89	eqcomd 2744	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥 RingHom 𝑧) = (𝑥(Hom ‘(RingCat‘𝑈))𝑧))
91	90	adantr 481	. . . . . . . 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 2842	. . . . . . 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 7284	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑧) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
95	94	adantl 482	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑧)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
96	58	adantr 481	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
97		simprr 770	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ 𝐶)
98		ovexd 7310	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥 RingHom 𝑧) ∈ V)
99	93, 95, 96, 97, 98	ovmpod 7425	. . . . . . . 8 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
100	99	adantr 481	. . . . . . 7 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
101	92, 100	eleqtrrd 2842	. . . . . 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 3123	. . . . 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 3123	. . . 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 512	. . 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 3103	. 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 17551	. 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 710	1 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ⟨cop 4567   class class class wbr 5074   I cid 5488   × cxp 5587   ↾ cres 5591   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Basecbs 16912  Hom chom 16973  compcco 16974  Catccat 17373  Idccid 17374  Hom_f chomf 17375   ⊆_cat cssc 17519  Subcatcsubc 17521  Ringcrg 19783   RingHom crh 19956  RingCatcringc 45561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-hom 16986  df-cco 16987  df-0g 17152  df-cat 17377  df-cid 17378  df-homf 17379  df-ssc 17522  df-resc 17523  df-subc 17524  df-estrc 17839  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-grp 18580  df-ghm 18832  df-mgp 19721  df-ur 19738  df-ring 19785  df-rnghom 19959  df-ringc 45563

This theorem is referenced by:  sringcat  45635  crhmsubc  45636  drhmsubc  45638  fldhmsubc  45642