Theorem f1ocnvb 5446

Description: A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and 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 5445	. 2 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
2		f1ocnv 5445	. . 3 |- ( `' F : B -1-1-onto-> A -> `' `' F : A -1-1-onto-> B )
3		dfrel2 5054	. . . 4 |- ( Rel F <-> `' `' F = F )
4		f1oeq1 5421	. . . 4 |- ( `' `' F = F -> ( `' `' F : A -1-1-onto-> B <-> F : A -1-1-onto-> B ) )
5	3, 4	sylbi 120	. . 3 |- ( Rel F -> ( `' `' F : A -1-1-onto-> B <-> F : A -1-1-onto-> B ) )
6	2, 5	syl5ib 153	. 2 |- ( Rel F -> ( `' F : B -1-1-onto-> A -> F : A -1-1-onto-> B ) )
7	1, 6	impbid2 142	1 |- ( Rel F -> ( F : A -1-1-onto-> B <-> `' F : B -1-1-onto-> A ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   `'ccnv 4603   Rel wrel 4609   -1-1-onto->wf1o 5187

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195

This theorem is referenced by: (None)