Theorem dffn4 5351

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 2139	. . 3 |- ran F = ran F
2	1	biantru 300	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F = ran F ) )
3		df-fo 5129	. 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 1331   ran crn 4540   Fn wfn 5118   -onto->wfo 5121

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 1425  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-fo 5129

This theorem is referenced by:  funforn  5352  ffoss  5399  tposf2  6165  mapsn  6584  fifo  6868