Theorem f1odm 5379

Description: The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)

Assertion

Ref	Expression
f1odm	|- ( F : A -1-1-onto-> B -> dom F = A )

Proof of Theorem f1odm

Step	Hyp	Ref	Expression
1		f1ofn 5376	. 2 |- ( F : A -1-1-onto-> B -> F Fn A )
2		fndm 5230	. 2 |- ( F Fn A -> dom F = A )
3	1, 2	syl 14	1 |- ( F : A -1-1-onto-> B -> dom F = A )

Syntax hints:   -> wi 4   = wceq 1332   dom cdm 4547   Fn wfn 5126   -1-1-onto->wf1o 5130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106

This theorem depends on definitions:  df-bi 116  df-fn 5134  df-f 5135  df-f1 5136  df-f1o 5138

This theorem is referenced by:  f1imacnv  5392  f1opw2  5984  xpcomco  6728  mapen  6748  ssenen  6753  phplem4  6757  phplem4on  6769  dif1en  6781  fiintim  6825  caseinl  6984  caseinr  6985  ctssdccl  7004  fihasheqf1oi  10566  hashfacen  10611  fisumss  11193