Theorem funfn 5246

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 2177	. . 3 |- dom A = dom A
2	1	biantru 302	. 2 |- ( Fun A <-> ( Fun A /\ dom A = dom A ) )
3		df-fn 5219	. 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 1353   dom cdm 4626   Fun wfun 5210   Fn wfn 5211

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 1449  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-fn 5219

This theorem is referenced by:  funfnd  5247  funssxp  5385  funforn  5445  funbrfvb  5558  funopfvb  5559  ssimaex  5577  fvco  5586  eqfunfv  5618  fvimacnvi  5630  unpreima  5641  respreima  5644  elrnrexdm  5655  elrnrexdmb  5656  ffvresb  5679  funresdfunsnss  5719  resfunexg  5737  funex  5739  elunirn  5766  smores  6292  smores2  6294  tfrlem1  6308  funresdfunsndc  6506  fundmfibi  6937  resunimafz0  10810  fclim  11301