ringcbasALTV - Mathbox for Alexander van der Vekens

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Theorem ringcbasALTV 47470

Description: Set of objects of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
ringcbasALTV.c	⊢ 𝐶 = (RingCatALTV‘𝑈)
ringcbasALTV.b	⊢ 𝐵 = (Base‘𝐶)
ringcbasALTV.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)

**Assertion**
Ref	Expression
ringcbasALTV	⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))

Proof of Theorem ringcbasALTV Dummy variables 𝑓 𝑔 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringcbasALTV.c	. . 3 ⊢ 𝐶 = (RingCatALTV‘𝑈)
2		ringcbasALTV.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		eqidd 2726	. . 3 ⊢ (𝜑 → (𝑈 ∩ Ring) = (𝑈 ∩ Ring))
4		eqidd 2726	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦)) = (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦)))
5		eqidd 2726	. . 3 ⊢ (𝜑 → (𝑣 ∈ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)), 𝑧 ∈ (𝑈 ∩ Ring) ↦ (𝑓 ∈ ((2^nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1^st ‘𝑣) RingHom (2^nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))) = (𝑣 ∈ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)), 𝑧 ∈ (𝑈 ∩ Ring) ↦ (𝑓 ∈ ((2^nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1^st ‘𝑣) RingHom (2^nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))))
6	1, 2, 3, 4, 5	ringcvalALTV 47459	. 2 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), (𝑈 ∩ Ring)⟩, ⟨(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)), 𝑧 ∈ (𝑈 ∩ Ring) ↦ (𝑓 ∈ ((2^nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1^st ‘𝑣) RingHom (2^nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩})
7		catstr 17942	. 2 ⊢ {⟨(Base‘ndx), (𝑈 ∩ Ring)⟩, ⟨(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)), 𝑧 ∈ (𝑈 ∩ Ring) ↦ (𝑓 ∈ ((2^nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1^st ‘𝑣) RingHom (2^nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩} Struct ⟨1, ;15⟩
8		baseid 17177	. 2 ⊢ Base = Slot (Base‘ndx)
9		snsstp1 4816	. 2 ⊢ {⟨(Base‘ndx), (𝑈 ∩ Ring)⟩} ⊆ {⟨(Base‘ndx), (𝑈 ∩ Ring)⟩, ⟨(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)), 𝑧 ∈ (𝑈 ∩ Ring) ↦ (𝑓 ∈ ((2^nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1^st ‘𝑣) RingHom (2^nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}
10		inex1g 5315	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V)
11	2, 10	syl 17	. 2 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V)
12		ringcbasALTV.b	. 2 ⊢ 𝐵 = (Base‘𝐶)
13	6, 7, 8, 9, 11, 12	strfv3 17168	1 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∩ cin 3940 {ctp 4629 ⟨cop 4631 × cxp 5671 ∘ ccom 5677 ‘cfv 6543 (class class class)co 7413 ∈ cmpo 7415 1^st c1st 7985 2^nd c2nd 7986 1c1 11134 5c5 12295 cdc 12702 ndxcnx 17156 Basecbs 17174 Hom chom 17238 compcco 17239 Ringcrg 20172 RingHom crh 20407 RingCatALTVcringcALTV 47457

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 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-struct 17110 df-slot 17145 df-ndx 17157 df-base 17175 df-hom 17251 df-cco 17252 df-ringcALTV 47458

This theorem is referenced by: ringchomfvalALTV 47471 ringccofvalALTV 47474 ringccatidALTV 47476 ringcbasbasALTV 47482 funcringcsetclem7ALTV 47489 srhmsubcALTVlem1 47493 srhmsubcALTV 47495

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