Theorem f1dm 5603

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

Assertion

Ref	Expression
f1dm	|- ( F : A -1-1-> B -> dom F = A )

Proof of Theorem f1dm

Step	Hyp	Ref	Expression
1		f1fn 5600	. 2 |- ( F : A -1-1-> B -> F Fn A )
2		fndm 5480	. 2 |- ( F Fn A -> dom F = A )
3	1, 2	syl 14	1 |- ( F : A -1-1-> B -> dom F = A )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   dom cdm 4774   Fn wfn 5372   -1-1->wf1 5374

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This proof depends on definitions:  df-bi 117  df-fn 5380  df-f 5381  df-f1 5382

This theorem is used by:  fun11iun  5660  tposf12  6540  f1dmvrnfibi  7258  f1vrnfibi  7259  exmidfodomrlemim  7553  hmeoimaf1o  15417  uspgr1edc  16493  exmidsbthrlem  17079