Theorem dffn2 5339

Description: Any function is a mapping into _V. (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
dffn2	|- ( F Fn A <-> F : A --> _V )

Proof of Theorem dffn2

Step	Hyp	Ref	Expression
1		ssv 3164	. . 3 |- ran F C_ _V
2	1	biantru 300	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F C_ _V ) )
3		df-f 5192	. 2 |- ( F : A --> _V <-> ( F Fn A /\ ran F C_ _V ) )
4	2, 3	bitr4i 186	1 |- ( F Fn A <-> F : A --> _V )

Syntax hints:   /\ wa 103   <-> wb 104   _Vcvv 2726   C_ wss 3116   ran crn 4605   Fn wfn 5183   -->wf 5184

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728  df-in 3122  df-ss 3129  df-f 5192

This theorem is referenced by:  f1cnvcnv  5404  fcoconst  5656  fnressn  5671  1stcof  6131  2ndcof  6132  fnmpo  6170  tposfn  6241  tfrlemibfn  6296  tfr1onlembfn  6312  mptelixpg  6700