Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  srhmsubc Structured version   Visualization version   GIF version

Theorem srhmsubc 42401
Description: According to df-subc 16519, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16547 and subcss2 16550). 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 eleq1 2718 . . . . . . 7 (𝑟 = 𝑥 → (𝑟 ∈ Ring ↔ 𝑥 ∈ Ring))
3 srhmsubc.s . . . . . . 7 𝑟𝑆 𝑟 ∈ Ring
42, 3vtoclri 3314 . . . . . 6 (𝑥𝑆𝑥 ∈ Ring)
54ssriv 3640 . . . . 5 𝑆 ⊆ Ring
6 sslin 3872 . . . . 5 (𝑆 ⊆ Ring → (𝑈𝑆) ⊆ (𝑈 ∩ Ring))
75, 6mp1i 13 . . . 4 (𝑈𝑉 → (𝑈𝑆) ⊆ (𝑈 ∩ Ring))
81, 7syl5eqss 3682 . . 3 (𝑈𝑉𝐶 ⊆ (𝑈 ∩ Ring))
9 ssid 3657 . . . . . 6 (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦)
10 eqid 2651 . . . . . . 7 (RingCat‘𝑈) = (RingCat‘𝑈)
11 eqid 2651 . . . . . . 7 (Base‘(RingCat‘𝑈)) = (Base‘(RingCat‘𝑈))
12 simpl 472 . . . . . . 7 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑈𝑉)
13 eqid 2651 . . . . . . 7 (Hom ‘(RingCat‘𝑈)) = (Hom ‘(RingCat‘𝑈))
143, 1srhmsubclem2 42399 . . . . . . . 8 ((𝑈𝑉𝑥𝐶) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
1514adantrr 753 . . . . . . 7 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
163, 1srhmsubclem2 42399 . . . . . . . 8 ((𝑈𝑉𝑦𝐶) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
1716adantrl 752 . . . . . . 7 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
1810, 11, 12, 13, 15, 17ringchom 42338 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(Hom ‘(RingCat‘𝑈))𝑦) = (𝑥 RingHom 𝑦))
199, 18syl5sseqr 3687 . . . . 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 6699 . . . . . . 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 809 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
25 simprr 811 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
26 ovexd 6720 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 RingHom 𝑦) ∈ V)
2721, 23, 24, 25, 26ovmpt2d 6830 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦))
28 eqid 2651 . . . . . 6 (Homf ‘(RingCat‘𝑈)) = (Homf ‘(RingCat‘𝑈))
2928, 11, 13, 15, 17homfval 16399 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(Homf ‘(RingCat‘𝑈))𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
3019, 27, 293sstr4d 3681 . . . 4 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))
3130ralrimivva 3000 . . 3 (𝑈𝑉 → ∀𝑥𝐶𝑦𝐶 (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))
32 ovex 6718 . . . . . 6 (𝑟 RingHom 𝑠) ∈ V
3320, 32fnmpt2i 7284 . . . . 5 𝐽 Fn (𝐶 × 𝐶)
3433a1i 11 . . . 4 (𝑈𝑉𝐽 Fn (𝐶 × 𝐶))
3528, 11homffn 16400 . . . . 5 (Homf ‘(RingCat‘𝑈)) Fn ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈)))
36 id 22 . . . . . . . . 9 (𝑈𝑉𝑈𝑉)
3710, 11, 36ringcbas 42336 . . . . . . . 8 (𝑈𝑉 → (Base‘(RingCat‘𝑈)) = (𝑈 ∩ Ring))
3837eqcomd 2657 . . . . . . 7 (𝑈𝑉 → (𝑈 ∩ Ring) = (Base‘(RingCat‘𝑈)))
3938sqxpeqd 5175 . . . . . 6 (𝑈𝑉 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈))))
4039fneq2d 6020 . . . . 5 (𝑈𝑉 → ((Homf ‘(RingCat‘𝑈)) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) ↔ (Homf ‘(RingCat‘𝑈)) Fn ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈)))))
4135, 40mpbiri 248 . . . 4 (𝑈𝑉 → (Homf ‘(RingCat‘𝑈)) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
42 inex1g 4834 . . . 4 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
4334, 41, 42isssc 16527 . . 3 (𝑈𝑉 → (𝐽cat (Homf ‘(RingCat‘𝑈)) ↔ (𝐶 ⊆ (𝑈 ∩ Ring) ∧ ∀𝑥𝐶𝑦𝐶 (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))))
448, 31, 43mpbir2and 977 . 2 (𝑈𝑉𝐽cat (Homf ‘(RingCat‘𝑈)))
451elin2 3834 . . . . . . . 8 (𝑥𝐶 ↔ (𝑥𝑈𝑥𝑆))
464adantl 481 . . . . . . . 8 ((𝑥𝑈𝑥𝑆) → 𝑥 ∈ Ring)
4745, 46sylbi 207 . . . . . . 7 (𝑥𝐶𝑥 ∈ Ring)
4847adantl 481 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝑥 ∈ Ring)
49 eqid 2651 . . . . . . 7 (Base‘𝑥) = (Base‘𝑥)
5049idrhm 18779 . . . . . 6 (𝑥 ∈ Ring → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
5148, 50syl 17 . . . . 5 ((𝑈𝑉𝑥𝐶) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
52 eqid 2651 . . . . . 6 (Id‘(RingCat‘𝑈)) = (Id‘(RingCat‘𝑈))
53 simpl 472 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝑈𝑉)
5410, 11, 52, 53, 14, 49ringcid 42350 . . . . 5 ((𝑈𝑉𝑥𝐶) → ((Id‘(RingCat‘𝑈))‘𝑥) = ( I ↾ (Base‘𝑥)))
5520a1i 11 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠)))
56 oveq12 6699 . . . . . . 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 476 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝑥𝐶)
59 ovexd 6720 . . . . . 6 ((𝑈𝑉𝑥𝐶) → (𝑥 RingHom 𝑥) ∈ V)
6055, 57, 58, 58, 59ovmpt2d 6830 . . . . 5 ((𝑈𝑉𝑥𝐶) → (𝑥𝐽𝑥) = (𝑥 RingHom 𝑥))
6151, 54, 603eltr4d 2745 . . . 4 ((𝑈𝑉𝑥𝐶) → ((Id‘(RingCat‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥))
62 eqid 2651 . . . . . . . . 9 (comp‘(RingCat‘𝑈)) = (comp‘(RingCat‘𝑈))
6310ringccat 42349 . . . . . . . . . 10 (𝑈𝑉 → (RingCat‘𝑈) ∈ Cat)
6463ad3antrrr 766 . . . . . . . . 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 798 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
6867adantr 480 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
693, 1srhmsubclem2 42399 . . . . . . . . . . 11 ((𝑈𝑉𝑧𝐶) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
7069ad2ant2rl 800 . . . . . . . . . 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 472 . . . . . . . . . . . . . . . 16 ((𝑦𝐶𝑧𝐶) → 𝑦𝐶)
7458, 73anim12i 589 . . . . . . . . . . . . . . 15 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥𝐶𝑦𝐶))
7572, 74jca 553 . . . . . . . . . . . . . 14 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)))
763, 1, 20srhmsubclem3 42400 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐽𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
7775, 76syl 17 . . . . . . . . . . . . 13 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥𝐽𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
7877eleq2d 2716 . . . . . . . . . . . 12 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑓 ∈ (𝑥𝐽𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
7978biimpcd 239 . . . . . . . . . . 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 445 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
823, 1, 20srhmsubclem3 42400 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ (𝑦𝐶𝑧𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
8382adantlr 751 . . . . . . . . . . . . 13 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
8483eleq2d 2716 . . . . . . . . . . . 12 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) ↔ 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
8584biimpd 219 . . . . . . . . . . 11 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
8685adantld 482 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
8786imp 444 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
8811, 13, 62, 64, 66, 68, 71, 81, 87catcocl 16393 . . . . . . . 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 42338 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥(Hom ‘(RingCat‘𝑈))𝑧) = (𝑥 RingHom 𝑧))
9089eqcomd 2657 . . . . . . . . 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 2733 . . . . . . 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 6699 . . . . . . . . . 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 811 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑧𝐶)
98 ovexd 6720 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥 RingHom 𝑧) ∈ V)
9993, 95, 96, 97, 98ovmpt2d 6830 . . . . . . . 8 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
10099adantr 480 . . . . . . 7 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
10192, 100eleqtrrd 2733 . . . . . 6 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
102101ralrimivva 3000 . . . . 5 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
103102ralrimivva 3000 . . . 4 ((𝑈𝑉𝑥𝐶) → ∀𝑦𝐶𝑧𝐶𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
10461, 103jca 553 . . 3 ((𝑈𝑉𝑥𝐶) → (((Id‘(RingCat‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝐶𝑧𝐶𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
105104ralrimiva 2995 . 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 16543 . 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 977 1 (𝑈𝑉𝐽 ∈ (Subcat‘(RingCat‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cin 3606  wss 3607  cop 4216   class class class wbr 4685   I cid 5052   × cxp 5141  cres 5145   Fn wfn 5921  cfv 5926  (class class class)co 6690  cmpt2 6692  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Idccid 16373  Homf chomf 16374  cat cssc 16514  Subcatcsubc 16516  Ringcrg 18593   RingHom crh 18760  RingCatcringc 42328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-hom 16013  df-cco 16014  df-0g 16149  df-cat 16376  df-cid 16377  df-homf 16378  df-ssc 16517  df-resc 16518  df-subc 16519  df-estrc 16810  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-grp 17472  df-ghm 17705  df-mgp 18536  df-ur 18548  df-ring 18595  df-rnghom 18763  df-ringc 42330
This theorem is referenced by:  sringcat  42402  crhmsubc  42403  drhmsubc  42405  fldhmsubc  42409
  Copyright terms: Public domain W3C validator