Theorem dff1o3 5132

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 896	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( ( F Fn A /\ ran F = B ) /\ Fun `' F ) )
2		dff1o2 5131	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
3		df-fo 4908	. . 3 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4	3	anbi1i 431	. 2 |- ( ( F : A -onto-> B /\ Fun `' F ) <-> ( ( F Fn A /\ ran F = B ) /\ Fun `' F ) )
5	1, 2, 4	3bitr4i 201	1 |- ( F : A -1-1-onto-> B <-> ( F : A -onto-> B /\ Fun `' F ) )

Syntax hints:   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   `'ccnv 4344   ran crn 4346   Fun wfun 4896   Fn wfn 4897   -onto->wfo 4900   -1-1-onto->wf1o 4901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909

This theorem is referenced by:  f1ofo  5133  resdif  5148  f11o  5159  f1opw  5707  1stconst  5842  2ndconst  5843  f1o2ndf1  5849  ssdomg  6258  phplem4  6318  phplem4on  6329