Theorem f1odm 5446

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 5443	. 2 |- ( F : A -1-1-onto-> B -> F Fn A )
2		fndm 5297	. 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 1348   dom cdm 4611   Fn wfn 5193   -1-1-onto->wf1o 5197

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 5201  df-f 5202  df-f1 5203  df-f1o 5205

This theorem is referenced by:  f1imacnv  5459  f1opw2  6055  xpcomco  6804  mapen  6824  ssenen  6829  phplem4  6833  phplem4on  6845  dif1en  6857  fiintim  6906  caseinl  7068  caseinr  7069  ctssdccl  7088  fihasheqf1oi  10722  hashfacen  10771  fisumss  11355