Theorem dffn3 5206

Description: A function maps to its range. (Contributed by NM, 1-Sep-1999.)

Assertion

Ref	Expression
dffn3	|- ( F Fn A <-> F : A --> ran F )

Proof of Theorem dffn3

Step	Hyp	Ref	Expression
1		ssid 3059	. . 3 |- ran F C_ ran F
2	1	biantru 297	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F C_ ran F ) )
3		df-f 5053	. 2 |- ( F : A --> ran F <-> ( F Fn A /\ ran F C_ ran F ) )
4	2, 3	bitr4i 186	1 |- ( F Fn A <-> F : A --> ran F )

Syntax hints:   /\ wa 103   <-> wb 104   C_ wss 3013   ran crn 4468   Fn wfn 5044   -->wf 5045

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-11 1449  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-in 3019  df-ss 3026  df-f 5053

This theorem is referenced by:  fsn2  5510  fo2ndf  6030