Theorem funforn 6800

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 6567	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6799	. 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)
3	1, 2	bitri 278	1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

Syntax hints:   ↔ wb 209  dom cdm 5662  ran crn 5663  Fun wfun 6531   Fn wfn 6532  –onto→wfo 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-cleq 2761  df-fn 6540  df-fo 6543

This theorem is referenced by:  fimacnvinrn  7067  imacosupp  8204  ordtypelem8  9486  wdomima2g  9547  imadomg  10517  gruima  10786  oppglsm  19711  1stcrestlem  23577  dfac14  23743  qtoptop2  23824  fsupprnfi  32977  imadomfi  42658  rn1st  45879