Theorem dffn3 5493

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 3247	. . 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 5330	. 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 3200   ran crn 4726   Fn wfn 5321   -->wf 5322

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213  df-f 5330

This theorem is referenced by:  fsn2  5821  fo2ndf  6391