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Theorem srhmsubcALTV 41398
Description: According to df-subc 16396, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16424 and subcss2 16427). 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.) (New usage is discouraged.)
Hypotheses
Ref Expression
srhmsubcALTV.s 𝑟𝑆 𝑟 ∈ Ring
srhmsubcALTV.c 𝐶 = (𝑈𝑆)
srhmsubcALTV.j 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))
Assertion
Ref Expression
srhmsubcALTV (𝑈𝑉𝐽 ∈ (Subcat‘(RingCatALTV‘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 srhmsubcALTV
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 srhmsubcALTV.c . . . 4 𝐶 = (𝑈𝑆)
2 eleq1 2686 . . . . . . 7 (𝑟 = 𝑥 → (𝑟 ∈ Ring ↔ 𝑥 ∈ Ring))
3 srhmsubcALTV.s . . . . . . 7 𝑟𝑆 𝑟 ∈ Ring
42, 3vtoclri 3269 . . . . . 6 (𝑥𝑆𝑥 ∈ Ring)
54ssriv 3588 . . . . 5 𝑆 ⊆ Ring
6 sslin 3819 . . . . 5 (𝑆 ⊆ Ring → (𝑈𝑆) ⊆ (𝑈 ∩ Ring))
75, 6mp1i 13 . . . 4 (𝑈𝑉 → (𝑈𝑆) ⊆ (𝑈 ∩ Ring))
81, 7syl5eqss 3630 . . 3 (𝑈𝑉𝐶 ⊆ (𝑈 ∩ Ring))
9 ssid 3605 . . . . . 6 (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦)
10 eqid 2621 . . . . . . 7 (RingCatALTV‘𝑈) = (RingCatALTV‘𝑈)
11 eqid 2621 . . . . . . 7 (Base‘(RingCatALTV‘𝑈)) = (Base‘(RingCatALTV‘𝑈))
12 simpl 473 . . . . . . 7 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑈𝑉)
13 eqid 2621 . . . . . . 7 (Hom ‘(RingCatALTV‘𝑈)) = (Hom ‘(RingCatALTV‘𝑈))
143, 1srhmsubcALTVlem1 41396 . . . . . . . 8 ((𝑈𝑉𝑥𝐶) → 𝑥 ∈ (Base‘(RingCatALTV‘𝑈)))
1514adantrr 752 . . . . . . 7 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥 ∈ (Base‘(RingCatALTV‘𝑈)))
163, 1srhmsubcALTVlem1 41396 . . . . . . . 8 ((𝑈𝑉𝑦𝐶) → 𝑦 ∈ (Base‘(RingCatALTV‘𝑈)))
1716adantrl 751 . . . . . . 7 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦 ∈ (Base‘(RingCatALTV‘𝑈)))
1810, 11, 12, 13, 15, 17ringchomALTV 41352 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦) = (𝑥 RingHom 𝑦))
199, 18syl5sseqr 3635 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 RingHom 𝑦) ⊆ (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦))
20 srhmsubcALTV.j . . . . . . 7 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))
2120a1i 11 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠)))
22 oveq12 6616 . . . . . . 7 ((𝑟 = 𝑥𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
2322adantl 482 . . . . . 6 (((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) ∧ (𝑟 = 𝑥𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
24 simprl 793 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
25 simprr 795 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
26 ovex 6635 . . . . . . 7 (𝑥 RingHom 𝑦) ∈ V
2726a1i 11 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 RingHom 𝑦) ∈ V)
2821, 23, 24, 25, 27ovmpt2d 6744 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦))
29 eqid 2621 . . . . . 6 (Homf ‘(RingCatALTV‘𝑈)) = (Homf ‘(RingCatALTV‘𝑈))
3029, 11, 13, 15, 17homfval 16276 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(Homf ‘(RingCatALTV‘𝑈))𝑦) = (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦))
3119, 28, 303sstr4d 3629 . . . 4 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCatALTV‘𝑈))𝑦))
3231ralrimivva 2965 . . 3 (𝑈𝑉 → ∀𝑥𝐶𝑦𝐶 (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCatALTV‘𝑈))𝑦))
33 ovex 6635 . . . . . 6 (𝑟 RingHom 𝑠) ∈ V
3420, 33fnmpt2i 7187 . . . . 5 𝐽 Fn (𝐶 × 𝐶)
3534a1i 11 . . . 4 (𝑈𝑉𝐽 Fn (𝐶 × 𝐶))
3629, 11homffn 16277 . . . . 5 (Homf ‘(RingCatALTV‘𝑈)) Fn ((Base‘(RingCatALTV‘𝑈)) × (Base‘(RingCatALTV‘𝑈)))
37 id 22 . . . . . . . . 9 (𝑈𝑉𝑈𝑉)
3810, 11, 37ringcbasALTV 41350 . . . . . . . 8 (𝑈𝑉 → (Base‘(RingCatALTV‘𝑈)) = (𝑈 ∩ Ring))
3938eqcomd 2627 . . . . . . 7 (𝑈𝑉 → (𝑈 ∩ Ring) = (Base‘(RingCatALTV‘𝑈)))
4039sqxpeqd 5103 . . . . . 6 (𝑈𝑉 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = ((Base‘(RingCatALTV‘𝑈)) × (Base‘(RingCatALTV‘𝑈))))
4140fneq2d 5942 . . . . 5 (𝑈𝑉 → ((Homf ‘(RingCatALTV‘𝑈)) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) ↔ (Homf ‘(RingCatALTV‘𝑈)) Fn ((Base‘(RingCatALTV‘𝑈)) × (Base‘(RingCatALTV‘𝑈)))))
4236, 41mpbiri 248 . . . 4 (𝑈𝑉 → (Homf ‘(RingCatALTV‘𝑈)) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
43 inex1g 4763 . . . 4 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
4435, 42, 43isssc 16404 . . 3 (𝑈𝑉 → (𝐽cat (Homf ‘(RingCatALTV‘𝑈)) ↔ (𝐶 ⊆ (𝑈 ∩ Ring) ∧ ∀𝑥𝐶𝑦𝐶 (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCatALTV‘𝑈))𝑦))))
458, 32, 44mpbir2and 956 . 2 (𝑈𝑉𝐽cat (Homf ‘(RingCatALTV‘𝑈)))
461elin2 3781 . . . . . . . 8 (𝑥𝐶 ↔ (𝑥𝑈𝑥𝑆))
474adantl 482 . . . . . . . 8 ((𝑥𝑈𝑥𝑆) → 𝑥 ∈ Ring)
4846, 47sylbi 207 . . . . . . 7 (𝑥𝐶𝑥 ∈ Ring)
4948adantl 482 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝑥 ∈ Ring)
50 eqid 2621 . . . . . . 7 (Base‘𝑥) = (Base‘𝑥)
5150idrhm 18655 . . . . . 6 (𝑥 ∈ Ring → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
5249, 51syl 17 . . . . 5 ((𝑈𝑉𝑥𝐶) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
53 eqid 2621 . . . . . 6 (Id‘(RingCatALTV‘𝑈)) = (Id‘(RingCatALTV‘𝑈))
54 simpl 473 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝑈𝑉)
5510, 11, 53, 54, 14, 50ringcidALTV 41358 . . . . 5 ((𝑈𝑉𝑥𝐶) → ((Id‘(RingCatALTV‘𝑈))‘𝑥) = ( I ↾ (Base‘𝑥)))
5620a1i 11 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠)))
57 oveq12 6616 . . . . . . 7 ((𝑟 = 𝑥𝑠 = 𝑥) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
5857adantl 482 . . . . . 6 (((𝑈𝑉𝑥𝐶) ∧ (𝑟 = 𝑥𝑠 = 𝑥)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
59 simpr 477 . . . . . 6 ((𝑈𝑉𝑥𝐶) → 𝑥𝐶)
60 ovex 6635 . . . . . . 7 (𝑥 RingHom 𝑥) ∈ V
6160a1i 11 . . . . . 6 ((𝑈𝑉𝑥𝐶) → (𝑥 RingHom 𝑥) ∈ V)
6256, 58, 59, 59, 61ovmpt2d 6744 . . . . 5 ((𝑈𝑉𝑥𝐶) → (𝑥𝐽𝑥) = (𝑥 RingHom 𝑥))
6352, 55, 623eltr4d 2713 . . . 4 ((𝑈𝑉𝑥𝐶) → ((Id‘(RingCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥))
64 eqid 2621 . . . . . . . . 9 (comp‘(RingCatALTV‘𝑈)) = (comp‘(RingCatALTV‘𝑈))
6510ringccatALTV 41357 . . . . . . . . . 10 (𝑈𝑉 → (RingCatALTV‘𝑈) ∈ Cat)
6665ad3antrrr 765 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (RingCatALTV‘𝑈) ∈ Cat)
6714adantr 481 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑥 ∈ (Base‘(RingCatALTV‘𝑈)))
6867adantr 481 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑥 ∈ (Base‘(RingCatALTV‘𝑈)))
6916ad2ant2r 782 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑦 ∈ (Base‘(RingCatALTV‘𝑈)))
7069adantr 481 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑦 ∈ (Base‘(RingCatALTV‘𝑈)))
713, 1srhmsubcALTVlem1 41396 . . . . . . . . . . 11 ((𝑈𝑉𝑧𝐶) → 𝑧 ∈ (Base‘(RingCatALTV‘𝑈)))
7271ad2ant2rl 784 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑧 ∈ (Base‘(RingCatALTV‘𝑈)))
7372adantr 481 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑧 ∈ (Base‘(RingCatALTV‘𝑈)))
7454adantr 481 . . . . . . . . . . . . . . 15 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑈𝑉)
75 simpl 473 . . . . . . . . . . . . . . . 16 ((𝑦𝐶𝑧𝐶) → 𝑦𝐶)
7659, 75anim12i 589 . . . . . . . . . . . . . . 15 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥𝐶𝑦𝐶))
7774, 76jca 554 . . . . . . . . . . . . . 14 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)))
783, 1, 20srhmsubcALTVlem2 41397 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐽𝑦) = (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦))
7977, 78syl 17 . . . . . . . . . . . . 13 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥𝐽𝑦) = (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦))
8079eleq2d 2684 . . . . . . . . . . . 12 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑓 ∈ (𝑥𝐽𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦)))
8180biimpcd 239 . . . . . . . . . . 11 (𝑓 ∈ (𝑥𝐽𝑦) → (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦)))
8281adantr 481 . . . . . . . . . 10 ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦)))
8382impcom 446 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦))
843, 1, 20srhmsubcALTVlem2 41397 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ (𝑦𝐶𝑧𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCatALTV‘𝑈))𝑧))
8584adantlr 750 . . . . . . . . . . . . 13 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCatALTV‘𝑈))𝑧))
8685eleq2d 2684 . . . . . . . . . . . 12 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) ↔ 𝑔 ∈ (𝑦(Hom ‘(RingCatALTV‘𝑈))𝑧)))
8786biimpd 219 . . . . . . . . . . 11 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) → 𝑔 ∈ (𝑦(Hom ‘(RingCatALTV‘𝑈))𝑧)))
8887adantld 483 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑔 ∈ (𝑦(Hom ‘(RingCatALTV‘𝑈))𝑧)))
8988imp 445 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑔 ∈ (𝑦(Hom ‘(RingCatALTV‘𝑈))𝑧))
9011, 13, 64, 66, 68, 70, 73, 83, 89catcocl 16270 . . . . . . . 8 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑧))
9110, 11, 74, 13, 67, 72ringchomALTV 41352 . . . . . . . . . 10 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑧) = (𝑥 RingHom 𝑧))
9291eqcomd 2627 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥 RingHom 𝑧) = (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑧))
9392adantr 481 . . . . . . . 8 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑥 RingHom 𝑧) = (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑧))
9490, 93eleqtrrd 2701 . . . . . . 7 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥 RingHom 𝑧))
9520a1i 11 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠)))
96 oveq12 6616 . . . . . . . . . 10 ((𝑟 = 𝑥𝑠 = 𝑧) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
9796adantl 482 . . . . . . . . 9 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑟 = 𝑥𝑠 = 𝑧)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
9859adantr 481 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑥𝐶)
99 simprr 795 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → 𝑧𝐶)
100 ovex 6635 . . . . . . . . . 10 (𝑥 RingHom 𝑧) ∈ V
101100a1i 11 . . . . . . . . 9 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥 RingHom 𝑧) ∈ V)
10295, 97, 98, 99, 101ovmpt2d 6744 . . . . . . . 8 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
103102adantr 481 . . . . . . 7 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
10494, 103eleqtrrd 2701 . . . . . 6 ((((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
105104ralrimivva 2965 . . . . 5 (((𝑈𝑉𝑥𝐶) ∧ (𝑦𝐶𝑧𝐶)) → ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
106105ralrimivva 2965 . . . 4 ((𝑈𝑉𝑥𝐶) → ∀𝑦𝐶𝑧𝐶𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
10763, 106jca 554 . . 3 ((𝑈𝑉𝑥𝐶) → (((Id‘(RingCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝐶𝑧𝐶𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
108107ralrimiva 2960 . 2 (𝑈𝑉 → ∀𝑥𝐶 (((Id‘(RingCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝐶𝑧𝐶𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
10929, 53, 64, 65, 35issubc2 16420 . 2 (𝑈𝑉 → (𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)) ↔ (𝐽cat (Homf ‘(RingCatALTV‘𝑈)) ∧ ∀𝑥𝐶 (((Id‘(RingCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝐶𝑧𝐶𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
11045, 108, 109mpbir2and 956 1 (𝑈𝑉𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cin 3555  wss 3556  cop 4156   class class class wbr 4615   I cid 4986   × cxp 5074  cres 5078   Fn wfn 5844  cfv 5849  (class class class)co 6607  cmpt2 6609  Basecbs 15784  Hom chom 15876  compcco 15877  Catccat 16249  Idccid 16250  Homf chomf 16251  cat cssc 16391  Subcatcsubc 16393  Ringcrg 18471   RingHom crh 18636  RingCatALTVcringcALTV 41308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-er 7690  df-map 7807  df-pm 7808  df-ixp 7856  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-3 11027  df-4 11028  df-5 11029  df-6 11030  df-7 11031  df-8 11032  df-9 11033  df-n0 11240  df-z 11325  df-dec 11441  df-uz 11635  df-fz 12272  df-struct 15786  df-ndx 15787  df-slot 15788  df-base 15789  df-sets 15790  df-plusg 15878  df-hom 15890  df-cco 15891  df-0g 16026  df-cat 16253  df-cid 16254  df-homf 16255  df-ssc 16394  df-subc 16396  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-mhm 17259  df-grp 17349  df-ghm 17582  df-mgp 18414  df-ur 18426  df-ring 18473  df-rnghom 18639  df-ringcALTV 41310
This theorem is referenced by:  sringcatALTV  41399  crhmsubcALTV  41400  drhmsubcALTV  41402  fldhmsubcALTV  41406
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