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Theorem srhmsubc 20576
Description: According to df-subc 17768, the subcategories (Subcatβ€˜πΆ) of a category 𝐢 are subsets of the homomorphisms of 𝐢 (see subcssc 17799 and subcss2 17802). 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 βˆ€π‘Ÿ ∈ 𝑆 π‘Ÿ ∈ Ring
srhmsubc.c 𝐢 = (π‘ˆ ∩ 𝑆)
srhmsubc.j 𝐽 = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))
Assertion
Ref Expression
srhmsubc (π‘ˆ ∈ 𝑉 β†’ 𝐽 ∈ (Subcatβ€˜(RingCatβ€˜π‘ˆ)))
Distinct variable groups:   𝑆,π‘Ÿ   𝐢,π‘Ÿ,𝑠   π‘ˆ,π‘Ÿ,𝑠   𝑉,π‘Ÿ,𝑠
Allowed substitution hints:   𝑆(𝑠)   𝐽(𝑠,π‘Ÿ)

Proof of Theorem srhmsubc
Dummy variables 𝑓 𝑔 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 srhmsubc.c . . . 4 𝐢 = (π‘ˆ ∩ 𝑆)
2 eleq1w 2810 . . . . . . 7 (π‘Ÿ = π‘₯ β†’ (π‘Ÿ ∈ Ring ↔ π‘₯ ∈ Ring))
3 srhmsubc.s . . . . . . 7 βˆ€π‘Ÿ ∈ 𝑆 π‘Ÿ ∈ Ring
42, 3vtoclri 3570 . . . . . 6 (π‘₯ ∈ 𝑆 β†’ π‘₯ ∈ Ring)
54ssriv 3981 . . . . 5 𝑆 βŠ† Ring
6 sslin 4229 . . . . 5 (𝑆 βŠ† Ring β†’ (π‘ˆ ∩ 𝑆) βŠ† (π‘ˆ ∩ Ring))
75, 6mp1i 13 . . . 4 (π‘ˆ ∈ 𝑉 β†’ (π‘ˆ ∩ 𝑆) βŠ† (π‘ˆ ∩ Ring))
81, 7eqsstrid 4025 . . 3 (π‘ˆ ∈ 𝑉 β†’ 𝐢 βŠ† (π‘ˆ ∩ Ring))
9 ssid 3999 . . . . . 6 (π‘₯ RingHom 𝑦) βŠ† (π‘₯ RingHom 𝑦)
10 eqid 2726 . . . . . . 7 (RingCatβ€˜π‘ˆ) = (RingCatβ€˜π‘ˆ)
11 eqid 2726 . . . . . . 7 (Baseβ€˜(RingCatβ€˜π‘ˆ)) = (Baseβ€˜(RingCatβ€˜π‘ˆ))
12 simpl 482 . . . . . . 7 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ π‘ˆ ∈ 𝑉)
13 eqid 2726 . . . . . . 7 (Hom β€˜(RingCatβ€˜π‘ˆ)) = (Hom β€˜(RingCatβ€˜π‘ˆ))
143, 1srhmsubclem2 20574 . . . . . . . 8 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) β†’ π‘₯ ∈ (Baseβ€˜(RingCatβ€˜π‘ˆ)))
1514adantrr 714 . . . . . . 7 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ π‘₯ ∈ (Baseβ€˜(RingCatβ€˜π‘ˆ)))
163, 1srhmsubclem2 20574 . . . . . . . 8 ((π‘ˆ ∈ 𝑉 ∧ 𝑦 ∈ 𝐢) β†’ 𝑦 ∈ (Baseβ€˜(RingCatβ€˜π‘ˆ)))
1716adantrl 713 . . . . . . 7 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ 𝑦 ∈ (Baseβ€˜(RingCatβ€˜π‘ˆ)))
1810, 11, 12, 13, 15, 17ringchom 20548 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯(Hom β€˜(RingCatβ€˜π‘ˆ))𝑦) = (π‘₯ RingHom 𝑦))
199, 18sseqtrrid 4030 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯ RingHom 𝑦) βŠ† (π‘₯(Hom β€˜(RingCatβ€˜π‘ˆ))𝑦))
20 srhmsubc.j . . . . . . 7 𝐽 = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))
2120a1i 11 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ 𝐽 = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠)))
22 oveq12 7414 . . . . . . 7 ((π‘Ÿ = π‘₯ ∧ 𝑠 = 𝑦) β†’ (π‘Ÿ RingHom 𝑠) = (π‘₯ RingHom 𝑦))
2322adantl 481 . . . . . 6 (((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) ∧ (π‘Ÿ = π‘₯ ∧ 𝑠 = 𝑦)) β†’ (π‘Ÿ RingHom 𝑠) = (π‘₯ RingHom 𝑦))
24 simprl 768 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ π‘₯ ∈ 𝐢)
25 simprr 770 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ 𝑦 ∈ 𝐢)
26 ovexd 7440 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯ RingHom 𝑦) ∈ V)
2721, 23, 24, 25, 26ovmpod 7556 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯𝐽𝑦) = (π‘₯ RingHom 𝑦))
28 eqid 2726 . . . . . 6 (Homf β€˜(RingCatβ€˜π‘ˆ)) = (Homf β€˜(RingCatβ€˜π‘ˆ))
2928, 11, 13, 15, 17homfval 17645 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯(Homf β€˜(RingCatβ€˜π‘ˆ))𝑦) = (π‘₯(Hom β€˜(RingCatβ€˜π‘ˆ))𝑦))
3019, 27, 293sstr4d 4024 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯𝐽𝑦) βŠ† (π‘₯(Homf β€˜(RingCatβ€˜π‘ˆ))𝑦))
3130ralrimivva 3194 . . 3 (π‘ˆ ∈ 𝑉 β†’ βˆ€π‘₯ ∈ 𝐢 βˆ€π‘¦ ∈ 𝐢 (π‘₯𝐽𝑦) βŠ† (π‘₯(Homf β€˜(RingCatβ€˜π‘ˆ))𝑦))
32 ovex 7438 . . . . . 6 (π‘Ÿ RingHom 𝑠) ∈ V
3320, 32fnmpoi 8055 . . . . 5 𝐽 Fn (𝐢 Γ— 𝐢)
3433a1i 11 . . . 4 (π‘ˆ ∈ 𝑉 β†’ 𝐽 Fn (𝐢 Γ— 𝐢))
3528, 11homffn 17646 . . . . 5 (Homf β€˜(RingCatβ€˜π‘ˆ)) Fn ((Baseβ€˜(RingCatβ€˜π‘ˆ)) Γ— (Baseβ€˜(RingCatβ€˜π‘ˆ)))
36 id 22 . . . . . . . . 9 (π‘ˆ ∈ 𝑉 β†’ π‘ˆ ∈ 𝑉)
3710, 11, 36ringcbas 20546 . . . . . . . 8 (π‘ˆ ∈ 𝑉 β†’ (Baseβ€˜(RingCatβ€˜π‘ˆ)) = (π‘ˆ ∩ Ring))
3837eqcomd 2732 . . . . . . 7 (π‘ˆ ∈ 𝑉 β†’ (π‘ˆ ∩ Ring) = (Baseβ€˜(RingCatβ€˜π‘ˆ)))
3938sqxpeqd 5701 . . . . . 6 (π‘ˆ ∈ 𝑉 β†’ ((π‘ˆ ∩ Ring) Γ— (π‘ˆ ∩ Ring)) = ((Baseβ€˜(RingCatβ€˜π‘ˆ)) Γ— (Baseβ€˜(RingCatβ€˜π‘ˆ))))
4039fneq2d 6637 . . . . 5 (π‘ˆ ∈ 𝑉 β†’ ((Homf β€˜(RingCatβ€˜π‘ˆ)) Fn ((π‘ˆ ∩ Ring) Γ— (π‘ˆ ∩ Ring)) ↔ (Homf β€˜(RingCatβ€˜π‘ˆ)) Fn ((Baseβ€˜(RingCatβ€˜π‘ˆ)) Γ— (Baseβ€˜(RingCatβ€˜π‘ˆ)))))
4135, 40mpbiri 258 . . . 4 (π‘ˆ ∈ 𝑉 β†’ (Homf β€˜(RingCatβ€˜π‘ˆ)) Fn ((π‘ˆ ∩ Ring) Γ— (π‘ˆ ∩ Ring)))
42 inex1g 5312 . . . 4 (π‘ˆ ∈ 𝑉 β†’ (π‘ˆ ∩ Ring) ∈ V)
4334, 41, 42isssc 17776 . . 3 (π‘ˆ ∈ 𝑉 β†’ (𝐽 βŠ†cat (Homf β€˜(RingCatβ€˜π‘ˆ)) ↔ (𝐢 βŠ† (π‘ˆ ∩ Ring) ∧ βˆ€π‘₯ ∈ 𝐢 βˆ€π‘¦ ∈ 𝐢 (π‘₯𝐽𝑦) βŠ† (π‘₯(Homf β€˜(RingCatβ€˜π‘ˆ))𝑦))))
448, 31, 43mpbir2and 710 . 2 (π‘ˆ ∈ 𝑉 β†’ 𝐽 βŠ†cat (Homf β€˜(RingCatβ€˜π‘ˆ)))
451elin2 4192 . . . . . . . 8 (π‘₯ ∈ 𝐢 ↔ (π‘₯ ∈ π‘ˆ ∧ π‘₯ ∈ 𝑆))
464adantl 481 . . . . . . . 8 ((π‘₯ ∈ π‘ˆ ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ Ring)
4745, 46sylbi 216 . . . . . . 7 (π‘₯ ∈ 𝐢 β†’ π‘₯ ∈ Ring)
4847adantl 481 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) β†’ π‘₯ ∈ Ring)
49 eqid 2726 . . . . . . 7 (Baseβ€˜π‘₯) = (Baseβ€˜π‘₯)
5049idrhm 20392 . . . . . 6 (π‘₯ ∈ Ring β†’ ( I β†Ύ (Baseβ€˜π‘₯)) ∈ (π‘₯ RingHom π‘₯))
5148, 50syl 17 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) β†’ ( I β†Ύ (Baseβ€˜π‘₯)) ∈ (π‘₯ RingHom π‘₯))
52 eqid 2726 . . . . . 6 (Idβ€˜(RingCatβ€˜π‘ˆ)) = (Idβ€˜(RingCatβ€˜π‘ˆ))
53 simpl 482 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) β†’ π‘ˆ ∈ 𝑉)
5410, 11, 52, 53, 14, 49ringcid 20560 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) β†’ ((Idβ€˜(RingCatβ€˜π‘ˆ))β€˜π‘₯) = ( I β†Ύ (Baseβ€˜π‘₯)))
5520a1i 11 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) β†’ 𝐽 = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠)))
56 oveq12 7414 . . . . . . 7 ((π‘Ÿ = π‘₯ ∧ 𝑠 = π‘₯) β†’ (π‘Ÿ RingHom 𝑠) = (π‘₯ RingHom π‘₯))
5756adantl 481 . . . . . 6 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (π‘Ÿ = π‘₯ ∧ 𝑠 = π‘₯)) β†’ (π‘Ÿ RingHom 𝑠) = (π‘₯ RingHom π‘₯))
58 simpr 484 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) β†’ π‘₯ ∈ 𝐢)
59 ovexd 7440 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) β†’ (π‘₯ RingHom π‘₯) ∈ V)
6055, 57, 58, 58, 59ovmpod 7556 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) β†’ (π‘₯𝐽π‘₯) = (π‘₯ RingHom π‘₯))
6151, 54, 603eltr4d 2842 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) β†’ ((Idβ€˜(RingCatβ€˜π‘ˆ))β€˜π‘₯) ∈ (π‘₯𝐽π‘₯))
62 eqid 2726 . . . . . . . . 9 (compβ€˜(RingCatβ€˜π‘ˆ)) = (compβ€˜(RingCatβ€˜π‘ˆ))
6310ringccat 20559 . . . . . . . . . 10 (π‘ˆ ∈ 𝑉 β†’ (RingCatβ€˜π‘ˆ) ∈ Cat)
6463ad3antrrr 727 . . . . . . . . 9 ((((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) ∧ (𝑓 ∈ (π‘₯𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) β†’ (RingCatβ€˜π‘ˆ) ∈ Cat)
6514adantr 480 . . . . . . . . . 10 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ π‘₯ ∈ (Baseβ€˜(RingCatβ€˜π‘ˆ)))
6665adantr 480 . . . . . . . . 9 ((((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) ∧ (𝑓 ∈ (π‘₯𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) β†’ π‘₯ ∈ (Baseβ€˜(RingCatβ€˜π‘ˆ)))
6716ad2ant2r 744 . . . . . . . . . 10 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ 𝑦 ∈ (Baseβ€˜(RingCatβ€˜π‘ˆ)))
6867adantr 480 . . . . . . . . 9 ((((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) ∧ (𝑓 ∈ (π‘₯𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) β†’ 𝑦 ∈ (Baseβ€˜(RingCatβ€˜π‘ˆ)))
693, 1srhmsubclem2 20574 . . . . . . . . . . 11 ((π‘ˆ ∈ 𝑉 ∧ 𝑧 ∈ 𝐢) β†’ 𝑧 ∈ (Baseβ€˜(RingCatβ€˜π‘ˆ)))
7069ad2ant2rl 746 . . . . . . . . . 10 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ 𝑧 ∈ (Baseβ€˜(RingCatβ€˜π‘ˆ)))
7170adantr 480 . . . . . . . . 9 ((((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) ∧ (𝑓 ∈ (π‘₯𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) β†’ 𝑧 ∈ (Baseβ€˜(RingCatβ€˜π‘ˆ)))
7253adantr 480 . . . . . . . . . . . . . . 15 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ π‘ˆ ∈ 𝑉)
73 simpl 482 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢) β†’ 𝑦 ∈ 𝐢)
7458, 73anim12i 612 . . . . . . . . . . . . . . 15 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢))
7572, 74jca 511 . . . . . . . . . . . . . 14 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ (π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)))
763, 1, 20srhmsubclem3 20575 . . . . . . . . . . . . . 14 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯𝐽𝑦) = (π‘₯(Hom β€˜(RingCatβ€˜π‘ˆ))𝑦))
7775, 76syl 17 . . . . . . . . . . . . 13 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ (π‘₯𝐽𝑦) = (π‘₯(Hom β€˜(RingCatβ€˜π‘ˆ))𝑦))
7877eleq2d 2813 . . . . . . . . . . . 12 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ (𝑓 ∈ (π‘₯𝐽𝑦) ↔ 𝑓 ∈ (π‘₯(Hom β€˜(RingCatβ€˜π‘ˆ))𝑦)))
7978biimpcd 248 . . . . . . . . . . 11 (𝑓 ∈ (π‘₯𝐽𝑦) β†’ (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ 𝑓 ∈ (π‘₯(Hom β€˜(RingCatβ€˜π‘ˆ))𝑦)))
8079adantr 480 . . . . . . . . . 10 ((𝑓 ∈ (π‘₯𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) β†’ (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ 𝑓 ∈ (π‘₯(Hom β€˜(RingCatβ€˜π‘ˆ))𝑦)))
8180impcom 407 . . . . . . . . 9 ((((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) ∧ (𝑓 ∈ (π‘₯𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) β†’ 𝑓 ∈ (π‘₯(Hom β€˜(RingCatβ€˜π‘ˆ))𝑦))
823, 1, 20srhmsubclem3 20575 . . . . . . . . . . . . . 14 ((π‘ˆ ∈ 𝑉 ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ (𝑦𝐽𝑧) = (𝑦(Hom β€˜(RingCatβ€˜π‘ˆ))𝑧))
8382adantlr 712 . . . . . . . . . . . . 13 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ (𝑦𝐽𝑧) = (𝑦(Hom β€˜(RingCatβ€˜π‘ˆ))𝑧))
8483eleq2d 2813 . . . . . . . . . . . 12 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ (𝑔 ∈ (𝑦𝐽𝑧) ↔ 𝑔 ∈ (𝑦(Hom β€˜(RingCatβ€˜π‘ˆ))𝑧)))
8584biimpd 228 . . . . . . . . . . 11 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ (𝑔 ∈ (𝑦𝐽𝑧) β†’ 𝑔 ∈ (𝑦(Hom β€˜(RingCatβ€˜π‘ˆ))𝑧)))
8685adantld 490 . . . . . . . . . 10 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ ((𝑓 ∈ (π‘₯𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) β†’ 𝑔 ∈ (𝑦(Hom β€˜(RingCatβ€˜π‘ˆ))𝑧)))
8786imp 406 . . . . . . . . 9 ((((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) ∧ (𝑓 ∈ (π‘₯𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) β†’ 𝑔 ∈ (𝑦(Hom β€˜(RingCatβ€˜π‘ˆ))𝑧))
8811, 13, 62, 64, 66, 68, 71, 81, 87catcocl 17638 . . . . . . . 8 ((((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) ∧ (𝑓 ∈ (π‘₯𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) β†’ (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(RingCatβ€˜π‘ˆ))𝑧)𝑓) ∈ (π‘₯(Hom β€˜(RingCatβ€˜π‘ˆ))𝑧))
8910, 11, 72, 13, 65, 70ringchom 20548 . . . . . . . . . 10 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ (π‘₯(Hom β€˜(RingCatβ€˜π‘ˆ))𝑧) = (π‘₯ RingHom 𝑧))
9089eqcomd 2732 . . . . . . . . 9 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ (π‘₯ RingHom 𝑧) = (π‘₯(Hom β€˜(RingCatβ€˜π‘ˆ))𝑧))
9190adantr 480 . . . . . . . 8 ((((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) ∧ (𝑓 ∈ (π‘₯𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) β†’ (π‘₯ RingHom 𝑧) = (π‘₯(Hom β€˜(RingCatβ€˜π‘ˆ))𝑧))
9288, 91eleqtrrd 2830 . . . . . . 7 ((((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) ∧ (𝑓 ∈ (π‘₯𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) β†’ (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(RingCatβ€˜π‘ˆ))𝑧)𝑓) ∈ (π‘₯ RingHom 𝑧))
9320a1i 11 . . . . . . . . 9 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ 𝐽 = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠)))
94 oveq12 7414 . . . . . . . . . 10 ((π‘Ÿ = π‘₯ ∧ 𝑠 = 𝑧) β†’ (π‘Ÿ RingHom 𝑠) = (π‘₯ RingHom 𝑧))
9594adantl 481 . . . . . . . . 9 ((((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) ∧ (π‘Ÿ = π‘₯ ∧ 𝑠 = 𝑧)) β†’ (π‘Ÿ RingHom 𝑠) = (π‘₯ RingHom 𝑧))
9658adantr 480 . . . . . . . . 9 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ π‘₯ ∈ 𝐢)
97 simprr 770 . . . . . . . . 9 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ 𝑧 ∈ 𝐢)
98 ovexd 7440 . . . . . . . . 9 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ (π‘₯ RingHom 𝑧) ∈ V)
9993, 95, 96, 97, 98ovmpod 7556 . . . . . . . 8 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ (π‘₯𝐽𝑧) = (π‘₯ RingHom 𝑧))
10099adantr 480 . . . . . . 7 ((((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) ∧ (𝑓 ∈ (π‘₯𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) β†’ (π‘₯𝐽𝑧) = (π‘₯ RingHom 𝑧))
10192, 100eleqtrrd 2830 . . . . . 6 ((((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) ∧ (𝑓 ∈ (π‘₯𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) β†’ (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(RingCatβ€˜π‘ˆ))𝑧)𝑓) ∈ (π‘₯𝐽𝑧))
102101ralrimivva 3194 . . . . 5 (((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) ∧ (𝑦 ∈ 𝐢 ∧ 𝑧 ∈ 𝐢)) β†’ βˆ€π‘“ ∈ (π‘₯𝐽𝑦)βˆ€π‘” ∈ (𝑦𝐽𝑧)(𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(RingCatβ€˜π‘ˆ))𝑧)𝑓) ∈ (π‘₯𝐽𝑧))
103102ralrimivva 3194 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) β†’ βˆ€π‘¦ ∈ 𝐢 βˆ€π‘§ ∈ 𝐢 βˆ€π‘“ ∈ (π‘₯𝐽𝑦)βˆ€π‘” ∈ (𝑦𝐽𝑧)(𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(RingCatβ€˜π‘ˆ))𝑧)𝑓) ∈ (π‘₯𝐽𝑧))
10461, 103jca 511 . . 3 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐢) β†’ (((Idβ€˜(RingCatβ€˜π‘ˆ))β€˜π‘₯) ∈ (π‘₯𝐽π‘₯) ∧ βˆ€π‘¦ ∈ 𝐢 βˆ€π‘§ ∈ 𝐢 βˆ€π‘“ ∈ (π‘₯𝐽𝑦)βˆ€π‘” ∈ (𝑦𝐽𝑧)(𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(RingCatβ€˜π‘ˆ))𝑧)𝑓) ∈ (π‘₯𝐽𝑧)))
105104ralrimiva 3140 . 2 (π‘ˆ ∈ 𝑉 β†’ βˆ€π‘₯ ∈ 𝐢 (((Idβ€˜(RingCatβ€˜π‘ˆ))β€˜π‘₯) ∈ (π‘₯𝐽π‘₯) ∧ βˆ€π‘¦ ∈ 𝐢 βˆ€π‘§ ∈ 𝐢 βˆ€π‘“ ∈ (π‘₯𝐽𝑦)βˆ€π‘” ∈ (𝑦𝐽𝑧)(𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(RingCatβ€˜π‘ˆ))𝑧)𝑓) ∈ (π‘₯𝐽𝑧)))
10628, 52, 62, 63, 34issubc2 17795 . 2 (π‘ˆ ∈ 𝑉 β†’ (𝐽 ∈ (Subcatβ€˜(RingCatβ€˜π‘ˆ)) ↔ (𝐽 βŠ†cat (Homf β€˜(RingCatβ€˜π‘ˆ)) ∧ βˆ€π‘₯ ∈ 𝐢 (((Idβ€˜(RingCatβ€˜π‘ˆ))β€˜π‘₯) ∈ (π‘₯𝐽π‘₯) ∧ βˆ€π‘¦ ∈ 𝐢 βˆ€π‘§ ∈ 𝐢 βˆ€π‘“ ∈ (π‘₯𝐽𝑦)βˆ€π‘” ∈ (𝑦𝐽𝑧)(𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(RingCatβ€˜π‘ˆ))𝑧)𝑓) ∈ (π‘₯𝐽𝑧)))))
10744, 105, 106mpbir2and 710 1 (π‘ˆ ∈ 𝑉 β†’ 𝐽 ∈ (Subcatβ€˜(RingCatβ€˜π‘ˆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  Vcvv 3468   ∩ cin 3942   βŠ† wss 3943  βŸ¨cop 4629   class class class wbr 5141   I cid 5566   Γ— cxp 5667   β†Ύ cres 5671   Fn wfn 6532  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  Basecbs 17153  Hom chom 17217  compcco 17218  Catccat 17617  Idccid 17618  Homf chomf 17619   βŠ†cat cssc 17763  Subcatcsubc 17765  Ringcrg 20138   RingHom crh 20371  RingCatcringc 20541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13491  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-hom 17230  df-cco 17231  df-0g 17396  df-cat 17621  df-cid 17622  df-homf 17623  df-ssc 17766  df-resc 17767  df-subc 17768  df-estrc 18086  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mhm 18713  df-grp 18866  df-ghm 19139  df-mgp 20040  df-ur 20087  df-ring 20140  df-rhm 20374  df-ringc 20542
This theorem is referenced by:  sringcat  20577  crhmsubc  20578  drhmsubc  20632  fldhmsubc  20636
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