Theorem f1ocnvb 5633

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 5632	. 2 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
2		f1ocnv 5632	. . 3 |- ( `' F : B -1-1-onto-> A -> `' `' F : A -1-1-onto-> B )
3		dfrel2 5218	. . . 4 |- ( Rel F <-> `' `' F = F )
4		f1oeq1 5607	. . . 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 1398   `'ccnv 4753   Rel wrel 4759   -1-1-onto->wf1o 5356

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364

This theorem is referenced by: (None)