Theorem f1orn 5534

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 5529	. 2 |- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F /\ ran F = ran F ) )
2		eqid 2205	. . 3 |- ran F = ran F
3		df-3an 983	. . 3 |- ( ( F Fn A /\ Fun `' F /\ ran F = ran F ) <-> ( ( F Fn A /\ Fun `' F ) /\ ran F = ran F ) )
4	2, 3	mpbiran2 944	. 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 981   = wceq 1373   `'ccnv 4675   ran crn 4677   Fun wfun 5266   Fn wfn 5267   -1-1-onto->wf1o 5271

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279

This theorem is referenced by:  f1f1orn  5535