Theorem dff1o4 5134

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 5131	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
2		3anass 889	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( F Fn A /\ ( Fun `' F /\ ran F = B ) ) )
3		df-rn 4356	. . . . . 6 |- ran F = dom `' F
4	3	eqeq1i 2047	. . . . 5 |- ( ran F = B <-> dom `' F = B )
5	4	anbi2i 430	. . . 4 |- ( ( Fun `' F /\ ran F = B ) <-> ( Fun `' F /\ dom `' F = B ) )
6		df-fn 4905	. . . 4 |- ( `' F Fn B <-> ( Fun `' F /\ dom `' F = B ) )
7	5, 6	bitr4i 176	. . 3 |- ( ( Fun `' F /\ ran F = B ) <-> `' F Fn B )
8	7	anbi2i 430	. 2 |- ( ( F Fn A /\ ( Fun `' F /\ ran F = B ) ) <-> ( F Fn A /\ `' F Fn B ) )
9	1, 2, 8	3bitri 195	1 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )

Syntax hints:   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   `'ccnv 4344   dom cdm 4345   ran crn 4346   Fun wfun 4896   Fn wfn 4897   -1-1-onto->wf1o 4901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-rn 4356  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909

This theorem is referenced by:  f1ocnv  5139  f1oun  5146  f1o00  5161  f1oi  5164  f1osn  5166  f1ompt  5320  f1ofveu  5500  f1ocnvd  5702