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Theorem mvtval 29057
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 5872 . . . . 5  |-  ( t  =  T  ->  (mType `  t )  =  (mType `  T ) )
21rneqd 5240 . . . 4  |-  ( t  =  T  ->  ran  (mType `  t )  =  ran  (mType `  T
) )
3 df-mvt 29042 . . . 4  |- mVT  =  ( t  e.  _V  |->  ran  (mType `  t )
)
4 fvex 5882 . . . . 5  |-  (mType `  T )  e.  _V
54rnex 6733 . . . 4  |-  ran  (mType `  T )  e.  _V
62, 3, 5fvmpt 5956 . . 3  |-  ( T  e.  _V  ->  (mVT `  T )  =  ran  (mType `  T ) )
7 rn0 5264 . . . . 5  |-  ran  (/)  =  (/)
87eqcomi 2470 . . . 4  |-  (/)  =  ran  (/)
9 fvprc 5866 . . . 4  |-  ( -.  T  e.  _V  ->  (mVT
`  T )  =  (/) )
10 fvprc 5866 . . . . 5  |-  ( -.  T  e.  _V  ->  (mType `  T )  =  (/) )
1110rneqd 5240 . . . 4  |-  ( -.  T  e.  _V  ->  ran  (mType `  T )  =  ran  (/) )
128, 9, 113eqtr4a 2524 . . 3  |-  ( -.  T  e.  _V  ->  (mVT
`  T )  =  ran  (mType `  T
) )
136, 12pm2.61i 164 . 2  |-  (mVT `  T )  =  ran  (mType `  T )
14 mvtval.f . 2  |-  V  =  (mVT `  T )
15 mvtval.y . . 3  |-  Y  =  (mType `  T )
1615rneqi 5239 . 2  |-  ran  Y  =  ran  (mType `  T
)
1713, 14, 163eqtr4i 2496 1  |-  V  =  ran  Y
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   ran crn 5009   ` cfv 5594  mTypecmty 29019  mVTcmvt 29020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fv 5602  df-mvt 29042
This theorem is referenced by:  mtyf  29109  mvtss  29110
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