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Theorem ringchomfvalALTV 41839
 Description: Set of arrows of the category of rings (in a universe). (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringcbasALTV.c 𝐶 = (RingCatALTV‘𝑈)
ringcbasALTV.b 𝐵 = (Base‘𝐶)
ringcbasALTV.u (𝜑𝑈𝑉)
ringchomfvalALTV.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
ringchomfvalALTV (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 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 ringchomfvalALTV
Dummy variables 𝑓 𝑔 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringchomfvalALTV.h . . 3 𝐻 = (Hom ‘𝐶)
2 ringcbasALTV.c . . . . 5 𝐶 = (RingCatALTV‘𝑈)
3 ringcbasALTV.u . . . . 5 (𝜑𝑈𝑉)
4 ringcbasALTV.b . . . . . 6 𝐵 = (Base‘𝐶)
52, 4, 3ringcbasALTV 41838 . . . . 5 (𝜑𝐵 = (𝑈 ∩ Ring))
6 eqidd 2611 . . . . 5 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)))
7 eqidd 2611 . . . . 5 (𝜑 → (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RingHom 𝑧), 𝑔 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑓𝑔))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RingHom 𝑧), 𝑔 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑓𝑔))))
82, 3, 5, 6, 7ringcvalALTV 41799 . . . 4 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RingHom 𝑧), 𝑔 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑓𝑔)))⟩})
98fveq2d 6107 . . 3 (𝜑 → (Hom ‘𝐶) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RingHom 𝑧), 𝑔 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑓𝑔)))⟩}))
101, 9syl5eq 2656 . 2 (𝜑𝐻 = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RingHom 𝑧), 𝑔 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑓𝑔)))⟩}))
11 fvex 6113 . . . . 5 (Base‘𝐶) ∈ V
124, 11eqeltri 2684 . . . 4 𝐵 ∈ V
1312, 12mpt2ex 7136 . . 3 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)) ∈ V
14 catstr 16440 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RingHom 𝑧), 𝑔 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑓𝑔)))⟩} Struct ⟨1, 15⟩
15 homid 15898 . . . 4 Hom = Slot (Hom ‘ndx)
16 snsstp2 4288 . . . 4 {⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RingHom 𝑧), 𝑔 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑓𝑔)))⟩}
1714, 15, 16strfv 15735 . . 3 ((𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)) ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RingHom 𝑧), 𝑔 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑓𝑔)))⟩}))
1813, 17mp1i 13 . 2 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑓 ∈ ((2nd𝑣) RingHom 𝑧), 𝑔 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑓𝑔)))⟩}))
1910, 18eqtr4d 2647 1 (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {ctp 4129  ⟨cop 4131   × cxp 5036   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  1c1 9816  5c5 10950  ;cdc 11369  ndxcnx 15692  Basecbs 15695  Hom chom 15779  compcco 15780   RingHom crh 18535  RingCatALTVcringcALTV 41796 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-ringcALTV 41798 This theorem is referenced by:  ringchomALTV  41840  ringccofvalALTV  41842
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