Theorem ringcbasALTV 49216

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 2761	. . 3 ⊢ (𝜑 → (𝑈 ∩ Ring) = (𝑈 ∩ Ring))
4		eqidd 2761	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦)) = (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦)))
5		eqidd 2761	. . 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 49205	. 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 18050	. 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 17305	. 2 ⊢ Base = Slot (Base‘ndx)
9		snsstp1 4777	. 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 5282	. . 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 17297	1 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∩ cin 3898  {ctp 4588  ⟨cop 4590   × cxp 5653   ∘ ccom 5659  ‘cfv 6533  (class class class)co 7414   ∈ cmpo 7416  1^st c1st 7985  2^nd c2nd 7986  1c1 11126  5c5 12323  ;cdc 12737  ndxcnx 17286  Basecbs 17302  Hom chom 17354  compcco 17355  Ringcrg 20373   RingHom crh 20611  RingCatALTVcringcALTV 49203

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-er 8697  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-2 12328  df-3 12329  df-4 12330  df-5 12331  df-6 12332  df-7 12333  df-8 12334  df-9 12335  df-n0 12530  df-z 12617  df-dec 12738  df-uz 12889  df-fz 13563  df-struct 17240  df-slot 17275  df-ndx 17287  df-base 17303  df-hom 17367  df-cco 17368  df-ringcALTV 49204

This theorem is used by:  ringchomfvalALTV  49217  ringccofvalALTV  49220  ringccatidALTV  49222  ringcbasbasALTV  49228  funcringcsetclem7ALTV  49235  srhmsubcALTVlem1  49239  srhmsubcALTV  49241