Theorem dffn3 5436

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 3213	. . 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 5275	. 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 3166   ran crn 4676   Fn wfn 5266   -->wf 5267

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179  df-f 5275

This theorem is referenced by:  fsn2  5754  fo2ndf  6313