Theorem dff1o4 5154

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
dff1o4	|- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )

Proof of Theorem dff1o4

Step	Hyp	Ref	Expression
1		dff1o2 5151	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
2		3anass 923	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( F Fn A /\ ( Fun `' F /\ ran F = B ) ) )
3		df-rn 4374	. . . . . 6 |- ran F = dom `' F
4	3	eqeq1i 2088	. . . . 5 |- ( ran F = B <-> dom `' F = B )
5	4	anbi2i 444	. . . 4 |- ( ( Fun `' F /\ ran F = B ) <-> ( Fun `' F /\ dom `' F = B ) )
6		df-fn 4925	. . . 4 |- ( `' F Fn B <-> ( Fun `' F /\ dom `' F = B ) )
7	5, 6	bitr4i 185	. . 3 |- ( ( Fun `' F /\ ran F = B ) <-> `' F Fn B )
8	7	anbi2i 444	. 2 |- ( ( F Fn A /\ ( Fun `' F /\ ran F = B ) ) <-> ( F Fn A /\ `' F Fn B ) )
9	1, 2, 8	3bitri 204	1 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )

Syntax hints:   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   `'ccnv 4362   dom cdm 4363   ran crn 4364   Fun wfun 4916   Fn wfn 4917   -1-1-onto->wf1o 4921

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 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986  df-rn 4374  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929

This theorem is referenced by:  f1ocnv  5159  f1oun  5166  f1o00  5181  f1oi  5184  f1osn  5186  f1ompt  5341  f1ofveu  5520  f1ocnvd  5722  f1od2  5876