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Theorem srhmsubc 20572
Description: According to df-subc 17766, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17797 and subcss2 17800). 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 𝑟𝑆 𝑟 ∈ Ring
srhmsubc.c 𝐶 = (𝑈𝑆)
srhmsubc.j 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))
Assertion
Ref Expression
srhmsubc (𝑈𝑉𝐽 ∈ (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.
StepHypRef Expression
1 srhmsubc.c . . . 4 𝐶 = (𝑈𝑆)
2 eleq1w 2815 . . . . . . 7 (𝑟 = 𝑥 → (𝑟 ∈ Ring ↔ 𝑥 ∈ Ring))
3 srhmsubc.s . . . . . . 7 𝑟𝑆 𝑟 ∈ Ring
42, 3vtoclri 3576 . . . . . 6 (𝑥𝑆𝑥 ∈ Ring)
54ssriv 3986 . . . . 5 𝑆 ⊆ Ring
6 sslin 4234 . . . . 5 (𝑆 ⊆ Ring → (𝑈𝑆) ⊆ (𝑈 ∩ Ring))
75, 6mp1i 13 . . . 4 (𝑈𝑉 → (𝑈𝑆) ⊆ (𝑈 ∩ Ring))
81, 7eqsstrid 4030 . . 3 (𝑈𝑉𝐶 ⊆ (𝑈 ∩ Ring))
9 ssid 4004 . . . . . 6 (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦)
10 eqid 2731 . . . . . . 7 (RingCat‘𝑈) = (RingCat‘𝑈)
11 eqid 2731 . . . . . . 7 (Base‘(RingCat‘𝑈)) = (Base‘(RingCat‘𝑈))
12 simpl 482 . . . . . . 7 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑈𝑉)
13 eqid 2731 . . . . . . 7 (Hom ‘(RingCat‘𝑈)) = (Hom ‘(RingCat‘𝑈))
143, 1srhmsubclem2 20570 . . . . . . . 8 ((𝑈𝑉𝑥𝐶) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
1514adantrr 714 . . . . . . 7 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
163, 1srhmsubclem2 20570 . . . . . . . 8 ((𝑈𝑉𝑦𝐶) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
1716adantrl 713 . . . . . . 7 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
1810, 11, 12, 13, 15, 17ringchom 20544 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(Hom ‘(RingCat‘𝑈))𝑦) = (𝑥 RingHom 𝑦))
199, 18sseqtrrid 4035 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 RingHom 𝑦) ⊆ (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
20 srhmsubc.j . . . . . . 7 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))
2120a1i 11 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠)))
22 oveq12 7421 . . . . . . 7 ((𝑟 = 𝑥𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
2322adantl 481 . . . . . 6 (((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) ∧ (𝑟 = 𝑥𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
24 simprl 768 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
25 simprr 770 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
26 ovexd 7447 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 RingHom 𝑦) ∈ V)
2721, 23, 24, 25, 26ovmpod 7563 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦))
28 eqid 2731 . . . . . 6 (Homf ‘(RingCat‘𝑈)) = (Homf ‘(RingCat‘𝑈))
2928, 11, 13, 15, 17homfval 17643 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(Homf ‘(RingCat‘𝑈))𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
3019, 27, 293sstr4d 4029 . . . 4 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))
3130ralrimivva 3199 . . 3 (𝑈𝑉 → ∀𝑥𝐶𝑦𝐶 (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))
32 ovex 7445 . . . . . 6 (𝑟 RingHom 𝑠) ∈ V
3320, 32fnmpoi 8060 . . . . 5 𝐽 Fn (𝐶 × 𝐶)
3433a1i 11 . . . 4 (𝑈𝑉𝐽 Fn (𝐶 × 𝐶))
3528, 11homffn 17644 . . . . 5 (Homf ‘(RingCat‘𝑈)) Fn ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈)))
36 id 22 . . . . . . . . 9 (𝑈𝑉𝑈𝑉)
3710, 11, 36ringcbas 20542 . . . . . . . 8 (𝑈𝑉 → (Base‘(RingCat‘𝑈)) = (𝑈 ∩ Ring))
3837eqcomd 2737 . . . . . . 7 (𝑈𝑉 → (𝑈 ∩ Ring) = (Base‘(RingCat‘𝑈)))
3938sqxpeqd 5708 . . . . . 6 (𝑈𝑉 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈))))
4039fneq2d 6643 . . . . 5 (𝑈𝑉 → ((Homf ‘(RingCat‘𝑈)) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) ↔ (Homf ‘(RingCat‘𝑈)) Fn ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈)))))
4135, 40mpbiri 258 . . . 4 (𝑈𝑉 → (Homf ‘(RingCat‘𝑈)) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
42 inex1g 5319 . . . 4 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
4334, 41, 42isssc 17774 . . 3 (𝑈𝑉 → (𝐽cat (Homf ‘(RingCat‘𝑈)) ↔ (𝐶 ⊆ (𝑈 ∩ Ring) ∧ ∀𝑥𝐶𝑦𝐶 (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))))
448, 31, 43mpbir2and 710 . 2 (𝑈𝑉𝐽cat (Homf ‘(RingCat‘𝑈)))
451elin2 4197 . . . . . . . 8 (𝑥𝐶 ↔ (𝑥𝑈𝑥𝑆))
464adantl 481 . . . . . . . 8 ((𝑥𝑈𝑥𝑆) → 𝑥 ∈ Ring)
4745, 46sylbi 216 . . . . . . 7 (𝑥𝐶𝑥 ∈ Ring)
4847adantl 481 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝑥 ∈ Ring)
49 eqid 2731 . . . . . . 7 (Base‘𝑥) = (Base‘𝑥)
5049idrhm 20388 . . . . . 6 (𝑥 ∈ Ring → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
5148, 50syl 17 . . . . 5 ((𝑈𝑉𝑥𝐶) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
52 eqid 2731 . . . . . 6 (Id‘(RingCat‘𝑈)) = (Id‘(RingCat‘𝑈))
53 simpl 482 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝑈𝑉)
5410, 11, 52, 53, 14, 49ringcid 20556 . . . . 5 ((𝑈𝑉𝑥𝐶) → ((Id‘(RingCat‘𝑈))‘𝑥) = ( I ↾ (Base‘𝑥)))
5520a1i 11 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠)))
56 oveq12 7421 . . . . . . 7 ((𝑟 = 𝑥𝑠 = 𝑥) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
5756adantl 481 . . . . . 6 (((𝑈𝑉𝑥𝐶) ∧ (𝑟 = 𝑥𝑠 = 𝑥)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
58 simpr 484 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝑥𝐶)
59 ovexd 7447 . . . . . 6 ((𝑈𝑉𝑥𝐶) → (𝑥 RingHom 𝑥) ∈ V)
6055, 57, 58, 58, 59ovmpod 7563 . . . . 5 ((𝑈𝑉𝑥𝐶) → (𝑥𝐽𝑥) = (𝑥 RingHom 𝑥))
6151, 54, 603eltr4d 2847 . . . 4 ((𝑈𝑉𝑥𝐶) → ((Id‘(RingCat‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥))
62 eqid 2731 . . . . . . . . 9 (comp‘(RingCat‘𝑈)) = (comp‘(RingCat‘𝑈))
6310ringccat 20555 . . . . . . . . . 10 (𝑈𝑉 → (RingCat‘𝑈) ∈ Cat)
6463ad3antrrr 727 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (RingCat‘𝑈) ∈ Cat)
6514adantr 480 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
6665adantr 480 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
6716ad2ant2r 744 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
6867adantr 480 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
693, 1srhmsubclem2 20570 . . . . . . . . . . 11 ((𝑈𝑉𝑧𝐶) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
7069ad2ant2rl 746 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
7170adantr 480 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
7253adantr 480 . . . . . . . . . . . . . . 15 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑈𝑉)
73 simpl 482 . . . . . . . . . . . . . . . 16 ((𝑦𝐶𝑧𝐶) → 𝑦𝐶)
7458, 73anim12i 612 . . . . . . . . . . . . . . 15 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥𝐶𝑦𝐶))
7572, 74jca 511 . . . . . . . . . . . . . 14 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)))
763, 1, 20srhmsubclem3 20571 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐽𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
7775, 76syl 17 . . . . . . . . . . . . 13 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥𝐽𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
7877eleq2d 2818 . . . . . . . . . . . 12 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑓 ∈ (𝑥𝐽𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
7978biimpcd 248 . . . . . . . . . . 11 (𝑓 ∈ (𝑥𝐽𝑦) → (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
8079adantr 480 . . . . . . . . . 10 ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
8180impcom 407 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
823, 1, 20srhmsubclem3 20571 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ (𝑦𝐶𝑧𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
8382adantlr 712 . . . . . . . . . . . . 13 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
8483eleq2d 2818 . . . . . . . . . . . 12 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) ↔ 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
8584biimpd 228 . . . . . . . . . . 11 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
8685adantld 490 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
8786imp 406 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
8811, 13, 62, 64, 66, 68, 71, 81, 87catcocl 17636 . . . . . . . 8 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑧))
8910, 11, 72, 13, 65, 70ringchom 20544 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥(Hom ‘(RingCat‘𝑈))𝑧) = (𝑥 RingHom 𝑧))
9089eqcomd 2737 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥 RingHom 𝑧) = (𝑥(Hom ‘(RingCat‘𝑈))𝑧))
9190adantr 480 . . . . . . . 8 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑥 RingHom 𝑧) = (𝑥(Hom ‘(RingCat‘𝑈))𝑧))
9288, 91eleqtrrd 2835 . . . . . . 7 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥 RingHom 𝑧))
9320a1i 11 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠)))
94 oveq12 7421 . . . . . . . . . 10 ((𝑟 = 𝑥𝑠 = 𝑧) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
9594adantl 481 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑟 = 𝑥𝑠 = 𝑧)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
9658adantr 480 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑥𝐶)
97 simprr 770 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑧𝐶)
98 ovexd 7447 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥 RingHom 𝑧) ∈ V)
9993, 95, 96, 97, 98ovmpod 7563 . . . . . . . 8 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
10099adantr 480 . . . . . . 7 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
10192, 100eleqtrrd 2835 . . . . . 6 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
102101ralrimivva 3199 . . . . 5 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
103102ralrimivva 3199 . . . 4 ((𝑈𝑉𝑥𝐶) → ∀𝑦𝐶𝑧𝐶𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
10461, 103jca 511 . . 3 ((𝑈𝑉𝑥𝐶) → (((Id‘(RingCat‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝐶𝑧𝐶𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
105104ralrimiva 3145 . 2 (𝑈𝑉 → ∀𝑥𝐶 (((Id‘(RingCat‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝐶𝑧𝐶𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
10628, 52, 62, 63, 34issubc2 17793 . 2 (𝑈𝑉 → (𝐽 ∈ (Subcat‘(RingCat‘𝑈)) ↔ (𝐽cat (Homf ‘(RingCat‘𝑈)) ∧ ∀𝑥𝐶 (((Id‘(RingCat‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝐶𝑧𝐶𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
10744, 105, 106mpbir2and 710 1 (𝑈𝑉𝐽 ∈ (Subcat‘(RingCat‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wral 3060  Vcvv 3473  cin 3947  wss 3948  cop 4634   class class class wbr 5148   I cid 5573   × cxp 5674  cres 5678   Fn wfn 6538  cfv 6543  (class class class)co 7412  cmpo 7414  Basecbs 17151  Hom chom 17215  compcco 17216  Catccat 17615  Idccid 17616  Homf chomf 17617  cat cssc 17761  Subcatcsubc 17763  Ringcrg 20134   RingHom crh 20367  RingCatcringc 20537
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-map 8828  df-pm 8829  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-fz 13492  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-hom 17228  df-cco 17229  df-0g 17394  df-cat 17619  df-cid 17620  df-homf 17621  df-ssc 17764  df-resc 17765  df-subc 17766  df-estrc 18084  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-mhm 18711  df-grp 18864  df-ghm 19135  df-mgp 20036  df-ur 20083  df-ring 20136  df-rhm 20370  df-ringc 20538
This theorem is referenced by:  sringcat  20573  crhmsubc  20574  drhmsubc  20628  fldhmsubc  20632
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