Theorem funforn 6785

Description: A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)

Assertion

Ref	Expression
funforn	⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

Proof of Theorem funforn

Step	Hyp	Ref	Expression
1		funfn 6551	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6784	. 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)
3	1, 2	bitri 277	1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

Syntax hints:   ↔ wb 208  dom cdm 5647  ran crn 5648  Fun wfun 6515   Fn wfn 6516  –onto→wfo 6519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-cleq 2754  df-fn 6524  df-fo 6527

This theorem is referenced by:  fimacnvinrn  7052  imacosupp  8189  ordtypelem8  9473  wdomima2g  9534  imadomg  10491  gruima  10760  oppglsm  19682  1stcrestlem  23512  dfac14  23678  qtoptop2  23759  fsupprnfi  32894  imadomfi  42619  rn1st  45848