Theorem funfn 5136

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

Assertion

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

Proof of Theorem funfn

Step	Hyp	Ref	Expression
1		eqid 2353	. . 3 |- dom A = dom A
2	1	biantru 491	. 2 |- (Fun A <-> (Fun A /\ dom A = dom A))
3		df-fn 4790	. 2 |- (A Fn dom A <-> (Fun A /\ dom A = dom A))
4	2, 3	bitr4i 243	1 |- (Fun A <-> A Fn dom A)

Syntax hints:   <-> wb 176   /\ wa 358   = wceq 1642  dom cdm 4772  Fun wfun 4775   Fn wfn 4776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-cleq 2346  df-fn 4790

This theorem is referenced by:  funssxp  5233  f1funfun  5263  funforn  5276  funbrfvb  5360  fvco  5383  eqfunfv  5397  fvimacnvi  5402  unpreima  5408  inpreima  5409  respreima  5410  ffvresb  5431  fnfullfun  5858  fvfullfun  5864  sbthlem1  6203  sbthlem3  6205