Theorem f1dm 5606

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 5603	. 2 |- ( F : A -1-1-> B -> F Fn A )
2		fndm 5483	. 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 4776   Fn wfn 5375   -1-1->wf1 5377

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 5383  df-f 5384  df-f1 5385

This theorem is used by:  fun11iun  5663  tposf12  6544  f1dmvrnfibi  7263  f1vrnfibi  7264  exmidfodomrlemim  7560  hmeoimaf1o  15552  uspgr1edc  16698  exmidsbthrlem  17285