Theorem rngcbasALTV 49362

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 2762	. . 3 ⊢ (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng))
4		eqidd 2762	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∩ Rng), 𝑦 ∈ (𝑈 ∩ Rng) ↦ (𝑥 RngHom 𝑦)) = (𝑥 ∈ (𝑈 ∩ Rng), 𝑦 ∈ (𝑈 ∩ Rng) ↦ (𝑥 RngHom 𝑦)))
5		eqidd 2762	. . 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 49361	. 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 18135	. 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 17390	. 2 ⊢ Base = Slot (Base‘ndx)
9		snsstp1 4777	. 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 5279	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
11	2, 10	syl 18	. 2 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V)
12		rngcbasALTV.b	. 2 ⊢ 𝐵 = (Base‘𝐶)
13	6, 7, 8, 9, 11, 12	strfv3 17382	1 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∩ cin 3898  {ctp 4588  ⟨cop 4590   × cxp 5649   ∘ ccom 5655  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  1c1 11201  5c5 12400  ;cdc 12814  ndxcnx 17371  Basecbs 17387  Hom chom 17439  compcco 17440  Rngcrng 20374   RngHom crnghm 20664  RngCatALTVcrngcALTV 49359

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 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-slot 17360  df-ndx 17372  df-base 17388  df-hom 17452  df-cco 17453  df-rngcALTV 49360

This theorem is used by:  rngchomfvalALTV  49363  rngccofvalALTV  49366  rngccatidALTV  49368  rngchomrnghmresALTV  49375  rhmsubcALTVlem3  49379  rhmsubcALTVlem4  49380