Theorem dffn2 5475

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 3246	. . 3 |- ran F C_ _V
2	1	biantru 302	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F C_ _V ) )
3		df-f 5322	. 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 2799   C_ wss 3197   ran crn 4720   Fn wfn 5313   -->wf 5314

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801  df-in 3203  df-ss 3210  df-f 5322

This theorem is referenced by:  f1cnvcnv  5542  fcoconst  5806  fnressn  5825  1stcof  6309  2ndcof  6310  fnmpo  6348  tposfn  6419  tfrlemibfn  6474  tfr1onlembfn  6490  mptelixpg  6881