Theorem f1of1 5145

Description: A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)

Assertion

Ref	Expression
f1of1	|- ( F : A -1-1-onto-> B -> F : A -1-1-> B )

Proof of Theorem f1of1

Step	Hyp	Ref	Expression
1		df-f1o 4929	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2	1	simplbi 268	1 |- ( F : A -1-1-onto-> B -> F : A -1-1-> B )

Syntax hints:   -> wi 4   -1-1->wf1 4919   -onto->wfo 4920   -1-1-onto->wf1o 4921

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104

This theorem depends on definitions:  df-bi 115  df-f1o 4929

This theorem is referenced by:  f1of  5146  f1oresrab  5350  f1ocnvfvrneq  5442  isores3  5475  isoini2  5478  f1oiso  5485  f1opw2  5726  tposf12  5907  enssdom  6265  phplem4  6341  phplem4on  6353  fidceq  6354  en2eqpr  6380  isotilem  6419  negfi  10110