Theorem dffn2 5323

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 3150	. . 3 |- ran F C_ _V
2	1	biantru 300	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F C_ _V ) )
3		df-f 5176	. 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 2712   C_ wss 3102   ran crn 4589   Fn wfn 5167   -->wf 5168

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714  df-in 3108  df-ss 3115  df-f 5176

This theorem is referenced by:  f1cnvcnv  5388  fcoconst  5640  fnressn  5655  1stcof  6113  2ndcof  6114  fnmpo  6152  tposfn  6222  tfrlemibfn  6277  tfr1onlembfn  6293  mptelixpg  6681