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Theorem mvtval 32804
Description: The set of variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mvtval.f 𝑉 = (mVT‘𝑇)
mvtval.y 𝑌 = (mType‘𝑇)
Assertion
Ref Expression
mvtval 𝑉 = ran 𝑌

Proof of Theorem mvtval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6661 . . . . 5 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
21rneqd 5795 . . . 4 (𝑡 = 𝑇 → ran (mType‘𝑡) = ran (mType‘𝑇))
3 df-mvt 32789 . . . 4 mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
4 fvex 6674 . . . . 5 (mType‘𝑇) ∈ V
54rnex 7612 . . . 4 ran (mType‘𝑇) ∈ V
62, 3, 5fvmpt 6759 . . 3 (𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
7 rn0 5783 . . . . 5 ran ∅ = ∅
87eqcomi 2833 . . . 4 ∅ = ran ∅
9 fvprc 6654 . . . 4 𝑇 ∈ V → (mVT‘𝑇) = ∅)
10 fvprc 6654 . . . . 5 𝑇 ∈ V → (mType‘𝑇) = ∅)
1110rneqd 5795 . . . 4 𝑇 ∈ V → ran (mType‘𝑇) = ran ∅)
128, 9, 113eqtr4a 2885 . . 3 𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
136, 12pm2.61i 185 . 2 (mVT‘𝑇) = ran (mType‘𝑇)
14 mvtval.f . 2 𝑉 = (mVT‘𝑇)
15 mvtval.y . . 3 𝑌 = (mType‘𝑇)
1615rneqi 5794 . 2 ran 𝑌 = ran (mType‘𝑇)
1713, 14, 163eqtr4i 2857 1 𝑉 = ran 𝑌
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2115  Vcvv 3480  c0 4276  ran crn 5543  cfv 6343  mTypecmty 32766  mVTcmvt 32767
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-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  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-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fv 6351  df-mvt 32789
This theorem is referenced by:  mtyf  32856  mvtss  32857
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