Theorem dffn2 5515

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 3264	. . 3 |- ran F C_ _V
2	1	biantru 302	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F C_ _V ) )
3		df-f 5361	. 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 2815   C_ wss 3214   ran crn 4755   Fn wfn 5352   -->wf 5353

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-v 2817  df-in 3220  df-ss 3227  df-f 5361

This theorem is referenced by:  f1cnvcnv  5589  fcoconst  5853  fnressn  5875  1stcof  6370  2ndcof  6371  fnmpo  6411  tposfn  6517  tfrlemibfn  6572  tfr1onlembfn  6588  mptelixpg  6982