Theorem rngcbasALTV 48254

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 2730	. . 3 ⊢ (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng))
4		eqidd 2730	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∩ Rng), 𝑦 ∈ (𝑈 ∩ Rng) ↦ (𝑥 RngHom 𝑦)) = (𝑥 ∈ (𝑈 ∩ Rng), 𝑦 ∈ (𝑈 ∩ Rng) ↦ (𝑥 RngHom 𝑦)))
5		eqidd 2730	. . 3 ⊢ (𝜑 → (𝑣 ∈ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)), 𝑧 ∈ (𝑈 ∩ Rng) ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))) = (𝑣 ∈ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)), 𝑧 ∈ (𝑈 ∩ Rng) ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))))
6	1, 2, 3, 4, 5	rngcvalALTV 48253	. 2 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Rng), 𝑦 ∈ (𝑈 ∩ Rng) ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)), 𝑧 ∈ (𝑈 ∩ Rng) ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩})
7		catstr 17867	. 2 ⊢ {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Rng), 𝑦 ∈ (𝑈 ∩ Rng) ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)), 𝑧 ∈ (𝑈 ∩ Rng) ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩} Struct ⟨1, ;15⟩
8		baseid 17123	. 2 ⊢ Base = Slot (Base‘ndx)
9		snsstp1 4767	. 2 ⊢ {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩} ⊆ {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Rng), 𝑦 ∈ (𝑈 ∩ Rng) ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)), 𝑧 ∈ (𝑈 ∩ Rng) ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}
10		inex1g 5258	. . 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 17115	1 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ∩ cin 3902  {ctp 4581  ⟨cop 4583   × cxp 5617   ∘ ccom 5623  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  1^st c1st 7922  2^nd c2nd 7923  1c1 11010  5c5 12186  ;cdc 12591  ndxcnx 17104  Basecbs 17120  Hom chom 17172  compcco 17173  Rngcrng 20037   RngHom crnghm 20319  RngCatALTVcrngcALTV 48251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-hom 17185  df-cco 17186  df-rngcALTV 48252

This theorem is referenced by:  rngchomfvalALTV  48255  rngccofvalALTV  48258  rngccatidALTV  48260  rngchomrnghmresALTV  48267  rhmsubcALTVlem3  48271  rhmsubcALTVlem4  48272