Theorem dff1o4 5375

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 5372	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
2		3anass 966	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( F Fn A /\ ( Fun `' F /\ ran F = B ) ) )
3		df-rn 4550	. . . . . 6 |- ran F = dom `' F
4	3	eqeq1i 2147	. . . . 5 |- ( ran F = B <-> dom `' F = B )
5	4	anbi2i 452	. . . 4 |- ( ( Fun `' F /\ ran F = B ) <-> ( Fun `' F /\ dom `' F = B ) )
6		df-fn 5126	. . . 4 |- ( `' F Fn B <-> ( Fun `' F /\ dom `' F = B ) )
7	5, 6	bitr4i 186	. . 3 |- ( ( Fun `' F /\ ran F = B ) <-> `' F Fn B )
8	7	anbi2i 452	. 2 |- ( ( F Fn A /\ ( Fun `' F /\ ran F = B ) ) <-> ( F Fn A /\ `' F Fn B ) )
9	1, 2, 8	3bitri 205	1 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   `'ccnv 4538   dom cdm 4539   ran crn 4540   Fun wfun 5117   Fn wfn 5118   -1-1-onto->wf1o 5122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084  df-rn 4550  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130

This theorem is referenced by:  f1ocnv  5380  f1oun  5387  f1o00  5402  f1oi  5405  f1osn  5407  f1ompt  5571  f1ofveu  5762  f1ocnvd  5972  f1od2  6132  mapsnf1o2  6590  sbthlemi9  6853  hmeof1o2  12477