Theorem dffn3 5378

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 3177	. . 3 |- ran F C_ ran F
2	1	biantru 302	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F C_ ran F ) )
3		df-f 5222	. 2 |- ( F : A --> ran F <-> ( F Fn A /\ ran F C_ ran F ) )
4	2, 3	bitr4i 187	1 |- ( F Fn A <-> F : A --> ran F )

Syntax hints:   /\ wa 104   <-> wb 105   C_ wss 3131   ran crn 4629   Fn wfn 5213   -->wf 5214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144  df-f 5222

This theorem is referenced by:  fsn2  5692  fo2ndf  6230