Theorem dffn2 5349

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 3169	. . 3 |- ran F C_ _V
2	1	biantru 300	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F C_ _V ) )
3		df-f 5202	. 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 2730   C_ wss 3121   ran crn 4612   Fn wfn 5193   -->wf 5194

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732  df-in 3127  df-ss 3134  df-f 5202

This theorem is referenced by:  f1cnvcnv  5414  fcoconst  5667  fnressn  5682  1stcof  6142  2ndcof  6143  fnmpo  6181  tposfn  6252  tfrlemibfn  6307  tfr1onlembfn  6323  mptelixpg  6712