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Theorem elfvexd 5723
Description: If a function value is nonempty, its argument is a set. Deduction form of elfvex 5722. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
elfvexd.1  |-  ( ph  ->  A  e.  ( B `
 C ) )
Assertion
Ref Expression
elfvexd  |-  ( ph  ->  C  e.  _V )

Proof of Theorem elfvexd
StepHypRef Expression
1 elfvexd.1 . 2  |-  ( ph  ->  A  e.  ( B `
 C ) )
2 elfvex 5722 . 2  |-  ( A  e.  ( B `  C )  ->  C  e.  _V )
31, 2syl 16 1  |-  ( ph  ->  C  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756   _Vcvv 2977   ` cfv 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4426  ax-pow 4475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-dm 4855  df-iota 5386  df-fv 5431
This theorem is referenced by:  mrieqv2d  14582  mreexmrid  14586  mreexexlem3d  14589  mreexexlem4d  14590  mreexexd  14591  mreexdomd  14592  acsdomd  15356  isirred  16796  tgclb  18580  alexsublem  19621  cnextcn  19644  ustssel  19785  fmucnd  19872  trcfilu  19874  cfiluweak  19875  ucnextcn  19884  imasdsf1olem  19953  imasf1oxmet  19955  comet  20093  restmetu  20167  esumcvg  26540  mzpcl34  29072
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