Theorem ringcbasALTV 48954

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 2770	. . 3 ⊢ (𝜑 → (𝑈 ∩ Ring) = (𝑈 ∩ Ring))
4		eqidd 2770	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦)) = (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦)))
5		eqidd 2770	. . 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 48943	. 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 18017	. 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 17272	. 2 ⊢ Base = Slot (Base‘ndx)
9		snsstp1 4786	. 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 5290	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V)
11	2, 10	syl 18	. 2 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V)
12		ringcbasALTV.b	. 2 ⊢ 𝐵 = (Base‘𝐶)
13	6, 7, 8, 9, 11, 12	strfv3 17264	1 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912  {ctp 4598  ⟨cop 4600   × cxp 5660   ∘ ccom 5666  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  1^st c1st 7984  2^nd c2nd 7985  1c1 11101  5c5 12298  ;cdc 12711  ndxcnx 17253  Basecbs 17269  Hom chom 17321  compcco 17322  Ringcrg 20315   RingHom crh 20551  RingCatALTVcringcALTV 48941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-uz 12863  df-fz 13536  df-struct 17207  df-slot 17242  df-ndx 17254  df-base 17270  df-hom 17334  df-cco 17335  df-ringcALTV 48942

This theorem is referenced by:  ringchomfvalALTV  48955  ringccofvalALTV  48958  ringccatidALTV  48960  ringcbasbasALTV  48966  funcringcsetclem7ALTV  48973  srhmsubcALTVlem1  48977  srhmsubcALTV  48979