Theorem f1orn 6147

Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)

Assertion

Ref	Expression
f1orn	|- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F ) )

Proof of Theorem f1orn

Step	Hyp	Ref	Expression
1		dff1o2 6142	. 2 |- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F /\ ran F = ran F ) )
2		eqid 2622	. . 3 |- ran F = ran F
3		df-3an 1039	. . 3 |- ( ( F Fn A /\ Fun `' F /\ ran F = ran F ) <-> ( ( F Fn A /\ Fun `' F ) /\ ran F = ran F ) )
4	2, 3	mpbiran2 954	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = ran F ) <-> ( F Fn A /\ Fun `' F ) )
5	1, 4	bitri 264	1 |- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   `'ccnv 5113   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -1-1-onto->wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  f1f1orn  6148  infdifsn  8554  efopnlem2  24403