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Theorem rngchomfvalALTV 44471
 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 RngHomo 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 44470 . . . . 5 (𝜑𝐵 = (𝑈 ∩ Rng))
6 eqidd 2825 . . . . 5 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)))
7 eqidd 2825 . . . . 5 (𝜑 → (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑔 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑓𝑔))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑔 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑓𝑔))))
82, 3, 5, 6, 7rngcvalALTV 44448 . . . 4 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑔 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑓𝑔)))⟩})
98fveq2d 6662 . . 3 (𝜑 → (Hom ‘𝐶) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑔 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑓𝑔)))⟩}))
101, 9syl5eq 2871 . 2 (𝜑𝐻 = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑔 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑓𝑔)))⟩}))
114fvexi 6672 . . . 4 𝐵 ∈ V
1211, 11mpoex 7767 . . 3 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)) ∈ V
13 catstr 17223 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑔 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑓𝑔)))⟩} Struct ⟨1, 15⟩
14 homid 16684 . . . 4 Hom = Slot (Hom ‘ndx)
15 snsstp2 4734 . . . 4 {⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑔 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑓𝑔)))⟩}
1613, 14, 15strfv 16527 . . 3 ((𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)) ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑔 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑓𝑔)))⟩}))
1712, 16mp1i 13 . 2 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑔 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑓𝑔)))⟩}))
1810, 17eqtr4d 2862 1 (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  Vcvv 3480  {ctp 4553  ⟨cop 4555   × cxp 5540   ∘ ccom 5546  ‘cfv 6343  (class class class)co 7145   ∈ cmpo 7147  1st c1st 7677  2nd c2nd 7678  1c1 10530  5c5 11688  ;cdc 12091  ndxcnx 16476  Basecbs 16479  Hom chom 16572  compcco 16573   RngHomo crngh 44372  RngCatALTVcrngcALTV 44445 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7571  df-1st 7679  df-2nd 7680  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-oadd 8096  df-er 8279  df-en 8500  df-dom 8501  df-sdom 8502  df-fin 8503  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11693  df-3 11694  df-4 11695  df-5 11696  df-6 11697  df-7 11698  df-8 11699  df-9 11700  df-n0 11891  df-z 11975  df-dec 12092  df-uz 12237  df-fz 12891  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-hom 16585  df-cco 16586  df-rngcALTV 44447 This theorem is referenced by:  rngchomALTV  44472  rngccofvalALTV  44474  rngchomffvalALTV  44482
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