Theorem f1orn 5470

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 5465	. 2 |- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F /\ ran F = ran F ) )
2		eqid 2177	. . 3 |- ran F = ran F
3		df-3an 980	. . 3 |- ( ( F Fn A /\ Fun `' F /\ ran F = ran F ) <-> ( ( F Fn A /\ Fun `' F ) /\ ran F = ran F ) )
4	2, 3	mpbiran2 941	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = ran F ) <-> ( F Fn A /\ Fun `' F ) )
5	1, 4	bitri 184	1 |- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   `'ccnv 4624   ran crn 4626   Fun wfun 5209   Fn wfn 5210   -1-1-onto->wf1o 5214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222

This theorem is referenced by:  f1f1orn  5471