Theorem rngcbasALTV 46434

Description: Set of objects of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)

Hypotheses

Ref	Expression
rngcbasALTV.c	⊢ 𝐶 = (RngCatALTV‘𝑈)
rngcbasALTV.b	⊢ 𝐵 = (Base‘𝐶)
rngcbasALTV.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)

Assertion

Ref	Expression
rngcbasALTV	⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))

Proof of Theorem rngcbasALTV

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

Step	Hyp	Ref	Expression
1		rngcbasALTV.c	. . 3 ⊢ 𝐶 = (RngCatALTV‘𝑈)
2		rngcbasALTV.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		eqidd 2732	. . 3 ⊢ (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng))
4		eqidd 2732	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∩ Rng), 𝑦 ∈ (𝑈 ∩ Rng) ↦ (𝑥 RngHomo 𝑦)) = (𝑥 ∈ (𝑈 ∩ Rng), 𝑦 ∈ (𝑈 ∩ Rng) ↦ (𝑥 RngHomo 𝑦)))
5		eqidd 2732	. . 3 ⊢ (𝜑 → (𝑣 ∈ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)), 𝑧 ∈ (𝑈 ∩ Rng) ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))) = (𝑣 ∈ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)), 𝑧 ∈ (𝑈 ∩ Rng) ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))))
6	1, 2, 3, 4, 5	rngcvalALTV 46412	. 2 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Rng), 𝑦 ∈ (𝑈 ∩ Rng) ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)), 𝑧 ∈ (𝑈 ∩ Rng) ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩})
7		catstr 17874	. 2 ⊢ {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Rng), 𝑦 ∈ (𝑈 ∩ Rng) ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)), 𝑧 ∈ (𝑈 ∩ Rng) ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩} Struct ⟨1, ;15⟩
8		baseid 17112	. 2 ⊢ Base = Slot (Base‘ndx)
9		snsstp1 4796	. 2 ⊢ {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩} ⊆ {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Rng), 𝑦 ∈ (𝑈 ∩ Rng) ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)), 𝑧 ∈ (𝑈 ∩ Rng) ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}
10		inex1g 5296	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
11	2, 10	syl 17	. 2 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V)
12		rngcbasALTV.b	. 2 ⊢ 𝐵 = (Base‘𝐶)
13	6, 7, 8, 9, 11, 12	strfv3 17103	1 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3459   ∩ cin 3927  {ctp 4610  ⟨cop 4612   × cxp 5651   ∘ ccom 5657  ‘cfv 6516  (class class class)co 7377   ∈ cmpo 7379  1^st c1st 7939  2^nd c2nd 7940  1c1 11076  5c5 12235  ;cdc 12642  ndxcnx 17091  Basecbs 17109  Hom chom 17173  compcco 17174  Rngcrng 46325   RngHomo crngh 46336  RngCatALTVcrngcALTV 46409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-1st 7941  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8670  df-en 8906  df-dom 8907  df-sdom 8908  df-fin 8909  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-nn 12178  df-2 12240  df-3 12241  df-4 12242  df-5 12243  df-6 12244  df-7 12245  df-8 12246  df-9 12247  df-n0 12438  df-z 12524  df-dec 12643  df-uz 12788  df-fz 13450  df-struct 17045  df-slot 17080  df-ndx 17092  df-base 17110  df-hom 17186  df-cco 17187  df-rngcALTV 46411

This theorem is referenced by:  rngchomfvalALTV  46435  rngccofvalALTV  46438  rngccatidALTV  46440  rngchomrnghmresALTV  46447  rhmsubcALTVlem3  46557  rhmsubcALTVlem4  46558