Theorem dffn2 4990

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 2959	. . 3 |- ran F C_ _V
2	1	biantru 286	. 2 |- ( F Fn A <-> ( F Fn A /\ ran F C_ _V))
3		df-f 4849	. 2 |- ( F :A --> _V <-> ( F Fn A /\ ran F C_ _V))
4	2, 3	bitr4i 176	1 |- ( F Fn A <-> F :A --> _V)

Syntax hints:   /\ wa 97   <-> wb 98   _Vcvv 2551   C_ wss 2911   ran crn 4289   Fn wfn 4840   -->wf 4841

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553  df-in 2918  df-ss 2925  df-f 4849

This theorem is referenced by:  f1cnvcnv  5043  fcoconst  5277  fnressn  5292  1stcof  5732  2ndcof  5733  fnmpt2  5770  tposfn  5829  tfrlemibfn  5883