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 45116
 Description: According to df-subc 17154, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17182 and subcss2 17185). 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 2834 . . . . . . 7 (𝑟 = 𝑥 → (𝑟 ∈ Ring ↔ 𝑥 ∈ Ring))
3 srhmsubc.s . . . . . . 7 𝑟𝑆 𝑟 ∈ Ring
42, 3vtoclri 3505 . . . . . 6 (𝑥𝑆𝑥 ∈ Ring)
54ssriv 3898 . . . . 5 𝑆 ⊆ Ring
6 sslin 4141 . . . . 5 (𝑆 ⊆ Ring → (𝑈𝑆) ⊆ (𝑈 ∩ Ring))
75, 6mp1i 13 . . . 4 (𝑈𝑉 → (𝑈𝑆) ⊆ (𝑈 ∩ Ring))
81, 7eqsstrid 3942 . . 3 (𝑈𝑉𝐶 ⊆ (𝑈 ∩ Ring))
9 ssid 3916 . . . . . 6 (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦)
10 eqid 2758 . . . . . . 7 (RingCat‘𝑈) = (RingCat‘𝑈)
11 eqid 2758 . . . . . . 7 (Base‘(RingCat‘𝑈)) = (Base‘(RingCat‘𝑈))
12 simpl 486 . . . . . . 7 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑈𝑉)
13 eqid 2758 . . . . . . 7 (Hom ‘(RingCat‘𝑈)) = (Hom ‘(RingCat‘𝑈))
143, 1srhmsubclem2 45114 . . . . . . . 8 ((𝑈𝑉𝑥𝐶) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
1514adantrr 716 . . . . . . 7 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
163, 1srhmsubclem2 45114 . . . . . . . 8 ((𝑈𝑉𝑦𝐶) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
1716adantrl 715 . . . . . . 7 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
1810, 11, 12, 13, 15, 17ringchom 45053 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(Hom ‘(RingCat‘𝑈))𝑦) = (𝑥 RingHom 𝑦))
199, 18sseqtrrid 3947 . . . . 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 7165 . . . . . . 7 ((𝑟 = 𝑥𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
2322adantl 485 . . . . . 6 (((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) ∧ (𝑟 = 𝑥𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
24 simprl 770 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
25 simprr 772 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
26 ovexd 7191 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 RingHom 𝑦) ∈ V)
2721, 23, 24, 25, 26ovmpod 7303 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦))
28 eqid 2758 . . . . . 6 (Homf ‘(RingCat‘𝑈)) = (Homf ‘(RingCat‘𝑈))
2928, 11, 13, 15, 17homfval 17033 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(Homf ‘(RingCat‘𝑈))𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
3019, 27, 293sstr4d 3941 . . . 4 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))
3130ralrimivva 3120 . . 3 (𝑈𝑉 → ∀𝑥𝐶𝑦𝐶 (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))
32 ovex 7189 . . . . . 6 (𝑟 RingHom 𝑠) ∈ V
3320, 32fnmpoi 7778 . . . . 5 𝐽 Fn (𝐶 × 𝐶)
3433a1i 11 . . . 4 (𝑈𝑉𝐽 Fn (𝐶 × 𝐶))
3528, 11homffn 17034 . . . . 5 (Homf ‘(RingCat‘𝑈)) Fn ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈)))
36 id 22 . . . . . . . . 9 (𝑈𝑉𝑈𝑉)
3710, 11, 36ringcbas 45051 . . . . . . . 8 (𝑈𝑉 → (Base‘(RingCat‘𝑈)) = (𝑈 ∩ Ring))
3837eqcomd 2764 . . . . . . 7 (𝑈𝑉 → (𝑈 ∩ Ring) = (Base‘(RingCat‘𝑈)))
3938sqxpeqd 5560 . . . . . 6 (𝑈𝑉 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈))))
4039fneq2d 6433 . . . . 5 (𝑈𝑉 → ((Homf ‘(RingCat‘𝑈)) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) ↔ (Homf ‘(RingCat‘𝑈)) Fn ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈)))))
4135, 40mpbiri 261 . . . 4 (𝑈𝑉 → (Homf ‘(RingCat‘𝑈)) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
42 inex1g 5193 . . . 4 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
4334, 41, 42isssc 17162 . . 3 (𝑈𝑉 → (𝐽cat (Homf ‘(RingCat‘𝑈)) ↔ (𝐶 ⊆ (𝑈 ∩ Ring) ∧ ∀𝑥𝐶𝑦𝐶 (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCat‘𝑈))𝑦))))
448, 31, 43mpbir2and 712 . 2 (𝑈𝑉𝐽cat (Homf ‘(RingCat‘𝑈)))
451elin2 4104 . . . . . . . 8 (𝑥𝐶 ↔ (𝑥𝑈𝑥𝑆))
464adantl 485 . . . . . . . 8 ((𝑥𝑈𝑥𝑆) → 𝑥 ∈ Ring)
4745, 46sylbi 220 . . . . . . 7 (𝑥𝐶𝑥 ∈ Ring)
4847adantl 485 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝑥 ∈ Ring)
49 eqid 2758 . . . . . . 7 (Base‘𝑥) = (Base‘𝑥)
5049idrhm 19567 . . . . . 6 (𝑥 ∈ Ring → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
5148, 50syl 17 . . . . 5 ((𝑈𝑉𝑥𝐶) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
52 eqid 2758 . . . . . 6 (Id‘(RingCat‘𝑈)) = (Id‘(RingCat‘𝑈))
53 simpl 486 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝑈𝑉)
5410, 11, 52, 53, 14, 49ringcid 45065 . . . . 5 ((𝑈𝑉𝑥𝐶) → ((Id‘(RingCat‘𝑈))‘𝑥) = ( I ↾ (Base‘𝑥)))
5520a1i 11 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠)))
56 oveq12 7165 . . . . . . 7 ((𝑟 = 𝑥𝑠 = 𝑥) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
5756adantl 485 . . . . . 6 (((𝑈𝑉𝑥𝐶) ∧ (𝑟 = 𝑥𝑠 = 𝑥)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
58 simpr 488 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝑥𝐶)
59 ovexd 7191 . . . . . 6 ((𝑈𝑉𝑥𝐶) → (𝑥 RingHom 𝑥) ∈ V)
6055, 57, 58, 58, 59ovmpod 7303 . . . . 5 ((𝑈𝑉𝑥𝐶) → (𝑥𝐽𝑥) = (𝑥 RingHom 𝑥))
6151, 54, 603eltr4d 2867 . . . 4 ((𝑈𝑉𝑥𝐶) → ((Id‘(RingCat‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥))
62 eqid 2758 . . . . . . . . 9 (comp‘(RingCat‘𝑈)) = (comp‘(RingCat‘𝑈))
6310ringccat 45064 . . . . . . . . . 10 (𝑈𝑉 → (RingCat‘𝑈) ∈ Cat)
6463ad3antrrr 729 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (RingCat‘𝑈) ∈ Cat)
6514adantr 484 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
6665adantr 484 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑥 ∈ (Base‘(RingCat‘𝑈)))
6716ad2ant2r 746 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
6867adantr 484 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑦 ∈ (Base‘(RingCat‘𝑈)))
693, 1srhmsubclem2 45114 . . . . . . . . . . 11 ((𝑈𝑉𝑧𝐶) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
7069ad2ant2rl 748 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
7170adantr 484 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑧 ∈ (Base‘(RingCat‘𝑈)))
7253adantr 484 . . . . . . . . . . . . . . 15 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑈𝑉)
73 simpl 486 . . . . . . . . . . . . . . . 16 ((𝑦𝐶𝑧𝐶) → 𝑦𝐶)
7458, 73anim12i 615 . . . . . . . . . . . . . . 15 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥𝐶𝑦𝐶))
7572, 74jca 515 . . . . . . . . . . . . . 14 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)))
763, 1, 20srhmsubclem3 45115 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐽𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
7775, 76syl 17 . . . . . . . . . . . . 13 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥𝐽𝑦) = (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
7877eleq2d 2837 . . . . . . . . . . . 12 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑓 ∈ (𝑥𝐽𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
7978biimpcd 252 . . . . . . . . . . 11 (𝑓 ∈ (𝑥𝐽𝑦) → (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
8079adantr 484 . . . . . . . . . 10 ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦)))
8180impcom 411 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(RingCat‘𝑈))𝑦))
823, 1, 20srhmsubclem3 45115 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ (𝑦𝐶𝑧𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
8382adantlr 714 . . . . . . . . . . . . 13 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
8483eleq2d 2837 . . . . . . . . . . . 12 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) ↔ 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
8584biimpd 232 . . . . . . . . . . 11 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
8685adantld 494 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧)))
8786imp 410 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑔 ∈ (𝑦(Hom ‘(RingCat‘𝑈))𝑧))
8811, 13, 62, 64, 66, 68, 71, 81, 87catcocl 17027 . . . . . . . 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 45053 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥(Hom ‘(RingCat‘𝑈))𝑧) = (𝑥 RingHom 𝑧))
9089eqcomd 2764 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥 RingHom 𝑧) = (𝑥(Hom ‘(RingCat‘𝑈))𝑧))
9190adantr 484 . . . . . . . 8 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑥 RingHom 𝑧) = (𝑥(Hom ‘(RingCat‘𝑈))𝑧))
9288, 91eleqtrrd 2855 . . . . . . 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 7165 . . . . . . . . . 10 ((𝑟 = 𝑥𝑠 = 𝑧) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
9594adantl 485 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑟 = 𝑥𝑠 = 𝑧)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
9658adantr 484 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑥𝐶)
97 simprr 772 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑧𝐶)
98 ovexd 7191 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥 RingHom 𝑧) ∈ V)
9993, 95, 96, 97, 98ovmpod 7303 . . . . . . . 8 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
10099adantr 484 . . . . . . 7 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
10192, 100eleqtrrd 2855 . . . . . 6 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
102101ralrimivva 3120 . . . . 5 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
103102ralrimivva 3120 . . . 4 ((𝑈𝑉𝑥𝐶) → ∀𝑦𝐶𝑧𝐶𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
10461, 103jca 515 . . 3 ((𝑈𝑉𝑥𝐶) → (((Id‘(RingCat‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝐶𝑧𝐶𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
105104ralrimiva 3113 . 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 17178 . 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 712 1 (𝑈𝑉𝐽 ∈ (Subcat‘(RingCat‘𝑈)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   ∩ cin 3859   ⊆ wss 3860  ⟨cop 4531   class class class wbr 5036   I cid 5433   × cxp 5526   ↾ cres 5530   Fn wfn 6335  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  Basecbs 16554  Hom chom 16647  compcco 16648  Catccat 17006  Idccid 17007  Homf chomf 17008   ⊆cat cssc 17149  Subcatcsubc 17151  Ringcrg 19378   RingHom crh 19548  RingCatcringc 45043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-er 8305  df-map 8424  df-pm 8425  df-ixp 8493  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-2 11750  df-3 11751  df-4 11752  df-5 11753  df-6 11754  df-7 11755  df-8 11756  df-9 11757  df-n0 11948  df-z 12034  df-dec 12151  df-uz 12296  df-fz 12953  df-struct 16556  df-ndx 16557  df-slot 16558  df-base 16560  df-sets 16561  df-ress 16562  df-plusg 16649  df-hom 16660  df-cco 16661  df-0g 16786  df-cat 17010  df-cid 17011  df-homf 17012  df-ssc 17152  df-resc 17153  df-subc 17154  df-estrc 17452  df-mgm 17931  df-sgrp 17980  df-mnd 17991  df-mhm 18035  df-grp 18185  df-ghm 18436  df-mgp 19321  df-ur 19333  df-ring 19380  df-rnghom 19551  df-ringc 45045 This theorem is referenced by:  sringcat  45117  crhmsubc  45118  drhmsubc  45120  fldhmsubc  45124
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