Theorem funforn 6833

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

Assertion

Ref	Expression
funforn	⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴⟶ontoran 𝐴)

Proof of Theorem funforn

Step	Hyp	Ref	Expression
1		funfn 6598	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6832	. 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴⟶ontoran 𝐴)
3	1, 2	bitri 278	1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴⟶ontoran 𝐴)

This proof depends on syntax axioms:   ↔ wb 209  dom cdm 5658  ran crn 5659  Fun wfun 6560   Fn wfn 6561  ⟶ontowfo 6564

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-fn 6569  df-fo 6572

This theorem is used by:  fimacnvinrn  7103  imacosupp  8264  ordtypelem8  9556  wdomima2g  9617  imadomg  10652  gruima  10926  oppglsm  19907  1stcrestlem  23825  dfac14  23992  qtoptop2  24073  fsupprnfi  33382  imadomfi  43318  rn1st  46558