Theorem dffn4 5624

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 2238	. . 3 |- ran F = ran F
2	1	biantru 302	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F = ran F ) )
3		df-fo 5386	. 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 )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   ran crn 4777   Fn wfn 5375   -onto->wfo 5378

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1502  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-fo 5386

This theorem is used by:  funforn  5625  fimadmfo  5627  ffoss  5675  tposf2  6543  mapsn  6977  fifo  7320  quslem  13705  gausslemma2dlem1f1o  16396