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Theorem rngchomfvalALTV 42486
 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 42485 . . . . 5 (𝜑𝐵 = (𝑈 ∩ Rng))
6 eqidd 2753 . . . . 5 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)))
7 eqidd 2753 . . . . 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 42463 . . . 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 6348 . . 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 2798 . 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)))⟩}))
11 fvex 6354 . . . . 5 (Base‘𝐶) ∈ V
124, 11eqeltri 2827 . . . 4 𝐵 ∈ V
1312, 12mpt2ex 7407 . . 3 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)) ∈ V
14 catstr 16810 . . . 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⟩
15 homid 16269 . . . 4 Hom = Slot (Hom ‘ndx)
16 snsstp2 4485 . . . 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)))⟩}
1714, 15, 16strfv 16101 . . 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)))⟩}))
1813, 17mp1i 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)))⟩}))
1910, 18eqtr4d 2789 1 (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1624   ∈ wcel 2131  Vcvv 3332  {ctp 4317  ⟨cop 4319   × cxp 5256   ∘ ccom 5262  ‘cfv 6041  (class class class)co 6805   ↦ cmpt2 6807  1st c1st 7323  2nd c2nd 7324  1c1 10121  5c5 11257  ;cdc 11677  ndxcnx 16048  Basecbs 16051  Hom chom 16146  compcco 16147   RngHomo crngh 42387  RngCatALTVcrngcALTV 42460 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-3 11264  df-4 11265  df-5 11266  df-6 11267  df-7 11268  df-8 11269  df-9 11270  df-n0 11477  df-z 11562  df-dec 11678  df-uz 11872  df-fz 12512  df-struct 16053  df-ndx 16054  df-slot 16055  df-base 16057  df-hom 16160  df-cco 16161  df-rngcALTV 42462 This theorem is referenced by:  rngchomALTV  42487  rngccofvalALTV  42489  rngchomffvalALTV  42497
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