Theorem funfn 5387

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 2234	. . 3 ⊢ dom 𝐴 = dom 𝐴
2	1	biantru 302	. 2 ⊢ (Fun 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴))
3		df-fn 5360	. 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 1398  dom cdm 4754  Fun wfun 5351   Fn wfn 5352

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 1498  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227  df-fn 5360

This theorem is referenced by:  funfnd  5388  funssxp  5537  funforn  5602  funbrfvb  5722  funopfvb  5723  ssimaex  5743  fvco  5752  eqfunfv  5785  fvimacnvi  5797  unpreima  5807  respreima  5810  elrnrexdm  5821  elrnrexdmb  5822  ffvresb  5845  funiun  5864  funresdfunsnss  5892  resfunexg  5910  funex  5914  elunirn  5945  suppval1  6452  funsssuppss  6471  smores  6536  smores2  6538  tfrlem1  6552  funresdfunsndc  6752  fundmfibi  7218  resunimafz0  11223  fclim  12004  ausgrumgrien  16291