Theorem ringcbasALTV 48488

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 2735	. . 3 ⊢ (𝜑 → (𝑈 ∩ Ring) = (𝑈 ∩ Ring))
4		eqidd 2735	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦)) = (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦)))
5		eqidd 2735	. . 3 ⊢ (𝜑 → (𝑣 ∈ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)), 𝑧 ∈ (𝑈 ∩ Ring) ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))) = (𝑣 ∈ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)), 𝑧 ∈ (𝑈 ∩ Ring) ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))))
6	1, 2, 3, 4, 5	ringcvalALTV 48477	. 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 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩})
7		catstr 17882	. 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 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩} Struct ⟨1, ;15⟩
8		baseid 17137	. 2 ⊢ Base = Slot (Base‘ndx)
9		snsstp1 4770	. 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 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}
10		inex1g 5262	. . 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 17129	1 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ∩ cin 3898  {ctp 4582  ⟨cop 4584   × cxp 5620   ∘ ccom 5626  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930  1c1 11025  5c5 12201  ;cdc 12605  ndxcnx 17118  Basecbs 17134  Hom chom 17186  compcco 17187  Ringcrg 20166   RingHom crh 20403  RingCatALTVcringcALTV 48475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-z 12487  df-dec 12606  df-uz 12750  df-fz 13422  df-struct 17072  df-slot 17107  df-ndx 17119  df-base 17135  df-hom 17199  df-cco 17200  df-ringcALTV 48476

This theorem is referenced by:  ringchomfvalALTV  48489  ringccofvalALTV  48492  ringccatidALTV  48494  ringcbasbasALTV  48500  funcringcsetclem7ALTV  48507  srhmsubcALTVlem1  48511  srhmsubcALTV  48513