Theorem funfn 5348

Description: An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)

Assertion

Ref	Expression
funfn	|- ( Fun A <-> A Fn dom A )

Proof of Theorem funfn

Step	Hyp	Ref	Expression
1		eqid 2229	. . 3 |- dom A = dom A
2	1	biantru 302	. 2 |- ( Fun A <-> ( Fun A /\ dom A = dom A ) )
3		df-fn 5321	. 2 |- ( A Fn dom A <-> ( Fun A /\ dom A = dom A ) )
4	2, 3	bitr4i 187	1 |- ( Fun A <-> A Fn dom A )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   dom cdm 4719   Fun wfun 5312   Fn wfn 5313

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-gen 1495  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-fn 5321

This theorem is referenced by:  funfnd  5349  funssxp  5493  funforn  5555  funbrfvb  5674  funopfvb  5675  ssimaex  5695  fvco  5704  eqfunfv  5737  fvimacnvi  5749  unpreima  5760  respreima  5763  elrnrexdm  5774  elrnrexdmb  5775  ffvresb  5798  funiun  5816  funresdfunsnss  5842  resfunexg  5860  funex  5862  elunirn  5890  smores  6438  smores2  6440  tfrlem1  6454  funresdfunsndc  6652  fundmfibi  7105  resunimafz0  11053  fclim  11805  ausgrumgrien  15968