Theorem dffn4 5416

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 2165	. . 3 |- ran F = ran F
2	1	biantru 300	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F = ran F ) )
3		df-fo 5194	. 2 |- ( F : A -onto-> ran F <-> ( F Fn A /\ ran F = ran F ) )
4	2, 3	bitr4i 186	1 |- ( F Fn A <-> F : A -onto-> ran F )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   ran crn 4605   Fn wfn 5183   -onto->wfo 5186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1437  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-fo 5194

This theorem is referenced by:  funforn  5417  ffoss  5464  tposf2  6236  mapsn  6656  fifo  6945