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  5495  funforn  5557  funbrfvb  5676  funopfvb  5677  ssimaex  5697  fvco  5706  eqfunfv  5739  fvimacnvi  5751  unpreima  5762  respreima  5765  elrnrexdm  5776  elrnrexdmb  5777  ffvresb  5800  funiun  5818  funresdfunsnss  5846  resfunexg  5864  funex  5866  elunirn  5896  smores  6444  smores2  6446  tfrlem1  6460  funresdfunsndc  6660  fundmfibi  7116  resunimafz0  11066  fclim  11820  ausgrumgrien  15983