Theorem funfn 5137

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 4791	. 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 4773  Fun wfun 4776   Fn wfn 4777

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 4791

This theorem is referenced by:  funssxp  5234  f1funfun  5264  funforn  5277  funbrfvb  5361  fvco  5384  eqfunfv  5398  fvimacnvi  5403  unpreima  5409  inpreima  5410  respreima  5411  ffvresb  5432  fnfullfun  5859  fvfullfun  5865  sbthlem1  6204  sbthlem3  6206