Theorem dffn4 5276

Description: A function maps onto its range. (Contributed by set.mm contributors, 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 2353	. . 3 |- ran F = ran F
2	1	biantru 491	. 2 |- (F Fn A <-> (F Fn A /\ ran F = ran F))
3		df-fo 4794	. 2 |- (F:A-onto->ran F <-> (F Fn A /\ ran F = ran F))
4	2, 3	bitr4i 243	1 |- (F Fn A <-> F:A-onto->ran F)

Syntax hints:   <-> wb 176   /\ wa 358   = wceq 1642  ran crn 4774   Fn wfn 4777  -onto->wfo 4780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-cleq 2346  df-fo 4794

This theorem is referenced by:  funforn  5277  ffoss  5315  mapsn  6027