Theorem dff1o4 5624

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 5621	. 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 4762	. . . . . 6 |- ran F = dom `' F
4	3	eqeq1i 2242	. . . . 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 5357	. . . 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 4750   dom cdm 4751   ran crn 4752   Fun wfun 5348   Fn wfn 5349   -1-1-onto->wf1o 5353

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 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3219  df-ss 3226  df-rn 4762  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361

This theorem is referenced by:  f1ocnv  5629  f1oun  5636  f1o00  5653  f1oi  5656  f1osn  5658  f1ompt  5830  f1ofveu  6040  f1ocnvd  6259  f1od2  6433  mapsnf1o2  6933  sbthlemi9  7237  xnn0nnen  10803  nninfctlemfo  12740  mhmf1o  13700  grpinvf1o  13800  ghmf1o  14009  rhmf1o  14330  hmeof1o2  15190