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Theorem elfvexd 5892
Description: If a function value is nonempty, its argument is a set. Deduction form of elfvex 5891. (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 5891 . 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 1767   _Vcvv 3113   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576  ax-pow 4625
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-dm 5009  df-iota 5549  df-fv 5594
This theorem is referenced by:  mrieqv2d  14890  mreexmrid  14894  mreexexlem3d  14897  mreexexlem4d  14898  mreexexd  14899  mreexdomd  14900  acsdomd  15664  telgsumfz  16810  isirred  17132  tgclb  19238  alexsublem  20279  cnextcn  20302  ustssel  20443  fmucnd  20530  trcfilu  20532  cfiluweak  20533  ucnextcn  20542  imasdsf1olem  20611  imasf1oxmet  20613  comet  20751  restmetu  20825  esumcvg  27732  mzpcl34  30267
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