Theorem f1odm 5477

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 5474	. 2 |- ( F : A -1-1-onto-> B -> F Fn A )
2		fndm 5327	. 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 1363   dom cdm 4638   Fn wfn 5223   -1-1-onto->wf1o 5227

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 5231  df-f 5232  df-f1 5233  df-f1o 5235

This theorem is referenced by:  f1imacnv  5490  f1opw2  6091  xpcomco  6840  mapen  6860  ssenen  6865  phplem4  6869  phplem4on  6881  dif1en  6893  fiintim  6942  caseinl  7104  caseinr  7105  ctssdccl  7124  fihasheqf1oi  10781  hashfacen  10830  fisumss  11414