Theorem dff1o4 5515

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 5512	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
2		3anass 984	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( F Fn A /\ ( Fun `' F /\ ran F = B ) ) )
3		df-rn 4675	. . . . . 6 |- ran F = dom `' F
4	3	eqeq1i 2204	. . . . 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 5262	. . . 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 980   = wceq 1364   `'ccnv 4663   dom cdm 4664   ran crn 4665   Fun wfun 5253   Fn wfn 5254   -1-1-onto->wf1o 5258

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170  df-rn 4675  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266

This theorem is referenced by:  f1ocnv  5520  f1oun  5527  f1o00  5542  f1oi  5545  f1osn  5547  f1ompt  5716  f1ofveu  5913  f1ocnvd  6129  f1od2  6302  mapsnf1o2  6764  sbthlemi9  7040  xnn0nnen  10546  nninfctlemfo  12232  mhmf1o  13172  grpinvf1o  13272  ghmf1o  13481  rhmf1o  13800  hmeof1o2  14628