Theorem dffn2 5530

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 3270	. . 3 |- ran F C_ _V
2	1	biantru 302	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F C_ _V ) )
3		df-f 5376	. 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 2821   C_ wss 3220   ran crn 4770   Fn wfn 5367   -->wf 5368

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823  df-in 3226  df-ss 3233  df-f 5376

This theorem is referenced by:  f1cnvcnv  5604  fcoconst  5870  fnressn  5892  1stcof  6387  2ndcof  6388  fnmpo  6428  tposfn  6534  tfrlemibfn  6589  tfr1onlembfn  6605  mptelixpg  7006