Theorem dffn4 5474

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

Step	Hyp	Ref	Expression
1		eqid 2193	. . 3 |- ran F = ran F
2	1	biantru 302	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F = ran F ) )
3		df-fo 5252	. 2 |- ( F : A -onto-> ran F <-> ( F Fn A /\ ran F = ran F ) )
4	2, 3	bitr4i 187	1 |- ( F Fn A <-> F : A -onto-> ran F )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   ran crn 4656   Fn wfn 5241   -onto->wfo 5244

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 5252

This theorem is referenced by:  funforn  5475  fimadmfo  5477  ffoss  5524  tposf2  6312  mapsn  6735  fifo  7029  quslem  12897  gausslemma2dlem1f1o  15124