Theorem dffn2 5274

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 3119	. . 3 |- ran F C_ _V
2	1	biantru 300	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F C_ _V ) )
3		df-f 5127	. 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 2686   C_ wss 3071   ran crn 4540   Fn wfn 5118   -->wf 5119

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 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688  df-in 3077  df-ss 3084  df-f 5127

This theorem is referenced by:  f1cnvcnv  5339  fcoconst  5591  fnressn  5606  1stcof  6061  2ndcof  6062  fnmpo  6100  tposfn  6170  tfrlemibfn  6225  tfr1onlembfn  6241  mptelixpg  6628