Theorem dffn3 5278

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 3112	. . 3 |- ran F C_ ran F
2	1	biantru 300	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F C_ ran F ) )
3		df-f 5122	. 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 3066   ran crn 4535   Fn wfn 5113   -->wf 5114

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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-in 3072  df-ss 3079  df-f 5122

This theorem is referenced by:  fsn2  5587  fo2ndf  6117