Theorem fnf 3628

Description: Any function is a mapping into V.

Assertion

Ref	Expression
fnf	|- (F Fn A <-> F:A-->V)

Proof of Theorem fnf

Step	Hyp	Ref	Expression
1		ssv 2081	. . 3 |- ran F (_ V
2		df-f 3194	. . . 4 |- (F:A-->V <-> (F Fn A /\ ran F (_ V))
3	2	biimpr 152	. . 3 |- ((F Fn A /\ ran F (_ V) -> F:A-->V)
4	1, 3	mpan2 696	. 2 |- (F Fn A -> F:A-->V)
5		ffn 3627	. 2 |- (F:A-->V -> F Fn A)
6	4, 5	impbi 157	1 |- (F Fn A <-> F:A-->V)

Syntax hints:   <-> wb 146   /\ wa 223  Vcvv 1811   (_ wss 2047  ran crn 3171   Fn wfn 3177  -->wf 3178

This theorem is referenced by:  f1cnv 3666  fnressn 3837  tz7.48lem 3955  1stcof 4101  fnoprab2g 4121  uzrdgfnuz 6306

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053  df-f 3194

Copyright terms: Public domain