Theorem f1ocnvb 5586

Description: A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and codomain/range interchanged. (Contributed by NM, 8-Dec-2003.)

Assertion

Ref	Expression
f1ocnvb	|- ( Rel F -> ( F : A -1-1-onto-> B <-> `' F : B -1-1-onto-> A ) )

Proof of Theorem f1ocnvb

Step	Hyp	Ref	Expression
1		f1ocnv 5585	. 2 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
2		f1ocnv 5585	. . 3 |- ( `' F : B -1-1-onto-> A -> `' `' F : A -1-1-onto-> B )
3		dfrel2 5179	. . . 4 |- ( Rel F <-> `' `' F = F )
4		f1oeq1 5560	. . . 4 |- ( `' `' F = F -> ( `' `' F : A -1-1-onto-> B <-> F : A -1-1-onto-> B ) )
5	3, 4	sylbi 121	. . 3 |- ( Rel F -> ( `' `' F : A -1-1-onto-> B <-> F : A -1-1-onto-> B ) )
6	2, 5	imbitrid 154	. 2 |- ( Rel F -> ( `' F : B -1-1-onto-> A -> F : A -1-1-onto-> B ) )
7	1, 6	impbid2 143	1 |- ( Rel F -> ( F : A -1-1-onto-> B <-> `' F : B -1-1-onto-> A ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   `'ccnv 4718   Rel wrel 4724   -1-1-onto->wf1o 5317

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325

This theorem is referenced by: (None)