Theorem dff1o4 5642

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 5639	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
2		3anass 1013	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( F Fn A /\ ( Fun `' F /\ ran F = B ) ) )
3		df-rn 4780	. . . . . 6 |- ran F = dom `' F
4	3	eqeq1i 2246	. . . . 5 |- ( ran F = B <-> dom `' F = B )
5	4	anbi2i 461	. . . 4 |- ( ( Fun `' F /\ ran F = B ) <-> ( Fun `' F /\ dom `' F = B ) )
6		df-fn 5375	. . . 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 461	. 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 1009   = wceq 1402   `'ccnv 4768   dom cdm 4769   ran crn 4770   Fun wfun 5366   Fn wfn 5367   -1-1-onto->wf1o 5371

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-rn 4780  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379

This theorem is referenced by:  f1ocnv  5647  f1oun  5654  f1o00  5671  f1oi  5674  f1osn  5676  f1ompt  5850  f1ofveu  6063  f1ocnvd  6282  f1o3d  6288  f1od2  6461  mapsnf1o2  6968  sbthlemi9  7272  xnn0nnen  10852  nninfctlemfo  12795  mhmf1o  13754  grpinvf1o  13852  ghmf1o  14055  rhmf1o  14448  hmeof1o2  15332