Theorem dff1o3 5157

Description: Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
dff1o3	|- ( F : A -1-1-onto-> B <-> ( F : A -onto-> B /\ Fun `' F ) )

Proof of Theorem dff1o3

Step	Hyp	Ref	Expression
1		3anan32 931	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( ( F Fn A /\ ran F = B ) /\ Fun `' F ) )
2		dff1o2 5156	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
3		df-fo 4932	. . 3 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4	3	anbi1i 446	. 2 |- ( ( F : A -onto-> B /\ Fun `' F ) <-> ( ( F Fn A /\ ran F = B ) /\ Fun `' F ) )
5	1, 2, 4	3bitr4i 210	1 |- ( F : A -1-1-onto-> B <-> ( F : A -onto-> B /\ Fun `' F ) )

Syntax hints:   /\ wa 102   <-> wb 103   /\ w3a 920   = wceq 1285   `'ccnv 4364   ran crn 4366   Fun wfun 4920   Fn wfn 4921   -onto->wfo 4924   -1-1-onto->wf1o 4925

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933

This theorem is referenced by:  f1ofo  5158  resdif  5173  f11o  5184  f1opw  5732  1stconst  5867  2ndconst  5868  f1o2ndf1  5874  ssdomg  6317  phplem4  6380  phplem4on  6392