Theorem f1odm 5467

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 5464	. 2 |- ( F : A -1-1-onto-> B -> F Fn A )
2		fndm 5317	. 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 1353   dom cdm 4628   Fn wfn 5213   -1-1-onto->wf1o 5217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This theorem depends on definitions:  df-bi 117  df-fn 5221  df-f 5222  df-f1 5223  df-f1o 5225

This theorem is referenced by:  f1imacnv  5480  f1opw2  6079  xpcomco  6828  mapen  6848  ssenen  6853  phplem4  6857  phplem4on  6869  dif1en  6881  fiintim  6930  caseinl  7092  caseinr  7093  ctssdccl  7112  fihasheqf1oi  10769  hashfacen  10818  fisumss  11402