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Theorem dffn4 5482
Description: A function maps onto its range. (Contributed by NM, 10-May-1998.)
Assertion
Ref Expression
dffn4  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )

Proof of Theorem dffn4
StepHypRef Expression
1 eqid 2193 . . 3  |-  ran  F  =  ran  F
21biantru 302 . 2  |-  ( F  Fn  A  <->  ( F  Fn  A  /\  ran  F  =  ran  F ) )
3 df-fo 5260 . 2  |-  ( F : A -onto-> ran  F  <->  ( F  Fn  A  /\  ran  F  =  ran  F
) )
42, 3bitr4i 187 1  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1364   ran crn 4660    Fn wfn 5249   -onto->wfo 5252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1460  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-fo 5260
This theorem is referenced by:  funforn  5483  fimadmfo  5485  ffoss  5532  tposf2  6321  mapsn  6744  fifo  7039  quslem  12907  gausslemma2dlem1f1o  15176
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