Theorem dff1o4 5165

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 5162	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
2		3anass 924	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( F Fn A /\ ( Fun `' F /\ ran F = B ) ) )
3		df-rn 4382	. . . . . 6 |- ran F = dom `' F
4	3	eqeq1i 2089	. . . . 5 |- ( ran F = B <-> dom `' F = B )
5	4	anbi2i 445	. . . 4 |- ( ( Fun `' F /\ ran F = B ) <-> ( Fun `' F /\ dom `' F = B ) )
6		df-fn 4935	. . . 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 445	. 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 920   = wceq 1285   `'ccnv 4370   dom cdm 4371   ran crn 4372   Fun wfun 4926   Fn wfn 4927   -1-1-onto->wf1o 4931

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-rn 4382  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939

This theorem is referenced by:  f1ocnv  5170  f1oun  5177  f1o00  5192  f1oi  5195  f1osn  5197  f1ompt  5352  f1ofveu  5531  f1ocnvd  5733  f1od2  5887