Theorem f1dm 5333

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 5330	. 2 |- ( F : A -1-1-> B -> F Fn A )
2		fndm 5222	. 2 |- ( F Fn A -> dom F = A )
3	1, 2	syl 14	1 |- ( F : A -1-1-> B -> dom F = A )

Syntax hints:   -> wi 4   = wceq 1331   dom cdm 4539   Fn wfn 5118   -1-1->wf1 5120

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 5126  df-f 5127  df-f1 5128

This theorem is referenced by:  fun11iun  5388  tposf12  6166  f1dmvrnfibi  6832  f1vrnfibi  6833  exmidfodomrlemim  7057  hmeoimaf1o  12483  exmidsbthrlem  13217