Theorem dff1o4 5468

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 5465	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
2		3anass 982	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( F Fn A /\ ( Fun `' F /\ ran F = B ) ) )
3		df-rn 4636	. . . . . 6 |- ran F = dom `' F
4	3	eqeq1i 2185	. . . . 5 |- ( ran F = B <-> dom `' F = B )
5	4	anbi2i 457	. . . 4 |- ( ( Fun `' F /\ ran F = B ) <-> ( Fun `' F /\ dom `' F = B ) )
6		df-fn 5218	. . . 4 |- ( `' F Fn B <-> ( Fun `' F /\ dom `' F = B ) )
7	5, 6	bitr4i 187	. . 3 |- ( ( Fun `' F /\ ran F = B ) <-> `' F Fn B )
8	7	anbi2i 457	. 2 |- ( ( F Fn A /\ ( Fun `' F /\ ran F = B ) ) <-> ( F Fn A /\ `' F Fn B ) )
9	1, 2, 8	3bitri 206	1 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   `'ccnv 4624   dom cdm 4625   ran crn 4626   Fun wfun 5209   Fn wfn 5210   -1-1-onto->wf1o 5214

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142  df-rn 4636  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222

This theorem is referenced by:  f1ocnv  5473  f1oun  5480  f1o00  5495  f1oi  5498  f1osn  5500  f1ompt  5666  f1ofveu  5860  f1ocnvd  6070  f1od2  6233  mapsnf1o2  6693  sbthlemi9  6961  mhmf1o  12793  grpinvf1o  12872  hmeof1o2  13679