Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mvtval Structured version   Visualization version   GIF version

Theorem mvtval 35110
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 6897 . . . . 5 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
21rneqd 5940 . . . 4 (𝑡 = 𝑇 → ran (mType‘𝑡) = ran (mType‘𝑇))
3 df-mvt 35095 . . . 4 mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
4 fvex 6910 . . . . 5 (mType‘𝑇) ∈ V
54rnex 7918 . . . 4 ran (mType‘𝑇) ∈ V
62, 3, 5fvmpt 7005 . . 3 (𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
7 rn0 5928 . . . . 5 ran ∅ = ∅
87eqcomi 2737 . . . 4 ∅ = ran ∅
9 fvprc 6889 . . . 4 𝑇 ∈ V → (mVT‘𝑇) = ∅)
10 fvprc 6889 . . . . 5 𝑇 ∈ V → (mType‘𝑇) = ∅)
1110rneqd 5940 . . . 4 𝑇 ∈ V → ran (mType‘𝑇) = ran ∅)
128, 9, 113eqtr4a 2794 . . 3 𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇))
136, 12pm2.61i 182 . 2 (mVT‘𝑇) = ran (mType‘𝑇)
14 mvtval.f . 2 𝑉 = (mVT‘𝑇)
15 mvtval.y . . 3 𝑌 = (mType‘𝑇)
1615rneqi 5939 . 2 ran 𝑌 = ran (mType‘𝑇)
1713, 14, 163eqtr4i 2766 1 𝑉 = ran 𝑌
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1534  wcel 2099  Vcvv 3471  c0 4323  ran crn 5679  cfv 6548  mTypecmty 35072  mVTcmvt 35073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fv 6556  df-mvt 35095
This theorem is referenced by:  mtyf  35162  mvtss  35163
  Copyright terms: Public domain W3C validator