Theorem dffn2 5447

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 3223	. . 3 |- ran F C_ _V
2	1	biantru 302	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F C_ _V ) )
3		df-f 5294	. 2 |- ( F : A --> _V <-> ( F Fn A /\ ran F C_ _V ) )
4	2, 3	bitr4i 187	1 |- ( F Fn A <-> F : A --> _V )

Syntax hints:   /\ wa 104   <-> wb 105   _Vcvv 2776   C_ wss 3174   ran crn 4694   Fn wfn 5285   -->wf 5286

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-v 2778  df-in 3180  df-ss 3187  df-f 5294

This theorem is referenced by:  f1cnvcnv  5514  fcoconst  5774  fnressn  5793  1stcof  6272  2ndcof  6273  fnmpo  6311  tposfn  6382  tfrlemibfn  6437  tfr1onlembfn  6453  mptelixpg  6844