Theorem dff1o4 5622

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 5619	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
2		3anass 1009	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( F Fn A /\ ( Fun `' F /\ ran F = B ) ) )
3		df-rn 4760	. . . . . 6 |- ran F = dom `' F
4	3	eqeq1i 2240	. . . . 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 5355	. . . 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 1005   = wceq 1398   `'ccnv 4748   dom cdm 4749   ran crn 4750   Fun wfun 5346   Fn wfn 5347   -1-1-onto->wf1o 5351

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-in 3217  df-ss 3224  df-rn 4760  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359

This theorem is referenced by:  f1ocnv  5627  f1oun  5634  f1o00  5651  f1oi  5654  f1osn  5656  f1ompt  5828  f1ofveu  6038  f1ocnvd  6257  f1od2  6431  mapsnf1o2  6931  sbthlemi9  7235  xnn0nnen  10799  nninfctlemfo  12736  mhmf1o  13683  grpinvf1o  13783  ghmf1o  13992  rhmf1o  14313  hmeof1o2  15173