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Theorem rngchomfvalALTV 47520
Description: Set of arrows 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 (𝜑𝑈𝑉)
rngchomfvalALTV.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
rngchomfvalALTV (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝑈   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem rngchomfvalALTV
Dummy variables 𝑓 𝑔 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rngchomfvalALTV.h . . 3 𝐻 = (Hom ‘𝐶)
2 rngcbasALTV.c . . . . 5 𝐶 = (RngCatALTV‘𝑈)
3 rngcbasALTV.u . . . . 5 (𝜑𝑈𝑉)
4 rngcbasALTV.b . . . . . 6 𝐵 = (Base‘𝐶)
52, 4, 3rngcbasALTV 47519 . . . . 5 (𝜑𝐵 = (𝑈 ∩ Rng))
6 eqidd 2726 . . . . 5 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦)))
7 eqidd 2726 . . . . 5 (𝜑 → (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHom 𝑧), 𝑔 ∈ ((1st𝑣) RngHom (2nd𝑣)) ↦ (𝑓𝑔))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHom 𝑧), 𝑔 ∈ ((1st𝑣) RngHom (2nd𝑣)) ↦ (𝑓𝑔))))
82, 3, 5, 6, 7rngcvalALTV 47518 . . . 4 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHom 𝑧), 𝑔 ∈ ((1st𝑣) RngHom (2nd𝑣)) ↦ (𝑓𝑔)))⟩})
98fveq2d 6900 . . 3 (𝜑 → (Hom ‘𝐶) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHom 𝑧), 𝑔 ∈ ((1st𝑣) RngHom (2nd𝑣)) ↦ (𝑓𝑔)))⟩}))
101, 9eqtrid 2777 . 2 (𝜑𝐻 = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHom 𝑧), 𝑔 ∈ ((1st𝑣) RngHom (2nd𝑣)) ↦ (𝑓𝑔)))⟩}))
114fvexi 6910 . . . 4 𝐵 ∈ V
1211, 11mpoex 8084 . . 3 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦)) ∈ V
13 catstr 17967 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHom 𝑧), 𝑔 ∈ ((1st𝑣) RngHom (2nd𝑣)) ↦ (𝑓𝑔)))⟩} Struct ⟨1, 15⟩
14 homid 17412 . . . 4 Hom = Slot (Hom ‘ndx)
15 snsstp2 4822 . . . 4 {⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHom 𝑧), 𝑔 ∈ ((1st𝑣) RngHom (2nd𝑣)) ↦ (𝑓𝑔)))⟩}
1613, 14, 15strfv 17192 . . 3 ((𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦)) ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦)) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHom 𝑧), 𝑔 ∈ ((1st𝑣) RngHom (2nd𝑣)) ↦ (𝑓𝑔)))⟩}))
1712, 16mp1i 13 . 2 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦)) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHom 𝑧), 𝑔 ∈ ((1st𝑣) RngHom (2nd𝑣)) ↦ (𝑓𝑔)))⟩}))
1810, 17eqtr4d 2768 1 (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3461  {ctp 4634  cop 4636   × cxp 5676  ccom 5682  cfv 6549  (class class class)co 7419  cmpo 7421  1st c1st 7992  2nd c2nd 7993  1c1 11146  5c5 12308  cdc 12715  ndxcnx 17181  Basecbs 17199  Hom chom 17263  compcco 17264   RngHom crnghm 20402  RngCatALTVcrngcALTV 47516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-struct 17135  df-slot 17170  df-ndx 17182  df-base 17200  df-hom 17276  df-cco 17277  df-rngcALTV 47517
This theorem is referenced by:  rngchomALTV  47521  rngccofvalALTV  47523  rngchomffvalALTV  47531
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