Theorem funforn 6808

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 6575	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6807	. 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)

This proof depends on syntax axioms:   ↔ wb 209  dom cdm 5651  ran crn 5652  Fun wfun 6537   Fn wfn 6538  –onto→wfo 6541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-fn 6546  df-fo 6549

This theorem is used by:  fimacnvinrn  7076  imacosupp  8234  ordtypelem8  9526  wdomima2g  9587  imadomg  10620  gruima  10894  oppglsm  19865  1stcrestlem  23779  dfac14  23946  qtoptop2  24027  fsupprnfi  33336  imadomfi  43189  rn1st  46421