Theorem f1odm 5587

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 5584	. 2 |- ( F : A -1-1-onto-> B -> F Fn A )
2		fndm 5429	. 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 1397   dom cdm 4725   Fn wfn 5321   -1-1-onto->wf1o 5325

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 5329  df-f 5330  df-f1 5331  df-f1o 5333

This theorem is referenced by:  f1imacnv  5600  f1opw2  6228  en2  6997  xpcomco  7009  mapen  7031  ssenen  7036  phplem4  7040  phplem4on  7053  dif1en  7067  fiintim  7122  caseinl  7289  caseinr  7290  ctssdccl  7309  fihasheqf1oi  11048  hashfacen  11099  fisumss  11952