Theorem rngcbasALTV 44328

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 2821	. . 3 ⊢ (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng))
4		eqidd 2821	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∩ Rng), 𝑦 ∈ (𝑈 ∩ Rng) ↦ (𝑥 RngHomo 𝑦)) = (𝑥 ∈ (𝑈 ∩ Rng), 𝑦 ∈ (𝑈 ∩ Rng) ↦ (𝑥 RngHomo 𝑦)))
5		eqidd 2821	. . 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 44306	. 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 17222	. 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 16538	. 2 ⊢ Base = Slot (Base‘ndx)
9		snsstp1 4742	. 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 5216	. . 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 16527	1 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113  Vcvv 3491   ∩ cin 3928  {ctp 4564  ⟨cop 4566   × cxp 5546   ∘ ccom 5552  ‘cfv 6348  (class class class)co 7149   ∈ cmpo 7151  1^st c1st 7680  2^nd c2nd 7681  1c1 10531  5c5 11689  ;cdc 12092  ndxcnx 16475  Basecbs 16478  Hom chom 16571  compcco 16572  Rngcrng 44219   RngHomo crngh 44230  RngCatALTVcrngcALTV 44303

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-nel 3123  df-ral 3142  df-rex 3143  df-reu 3144  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7574  df-1st 7682  df-2nd 7683  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-1o 8095  df-oadd 8099  df-er 8282  df-en 8503  df-dom 8504  df-sdom 8505  df-fin 8506  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11632  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12890  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-hom 16584  df-cco 16585  df-rngcALTV 44305

This theorem is referenced by:  rngchomfvalALTV  44329  rngccofvalALTV  44332  rngccatidALTV  44334  rngchomrnghmresALTV  44341  rhmsubcALTVlem3  44451  rhmsubcALTVlem4  44452