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Theorem cnvcnvres 5286
Description: The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
Assertion
Ref Expression
cnvcnvres  |-  `' `' ( A  |`  B )  =  ( `' `' A  |`  B )

Proof of Theorem cnvcnvres
StepHypRef Expression
1 relres 5120 . . 3  |-  Rel  ( A  |`  B )
2 dfrel2 5273 . . 3  |-  ( Rel  ( A  |`  B )  <->  `' `' ( A  |`  B )  =  ( A  |`  B )
)
31, 2mpbi 208 . 2  |-  `' `' ( A  |`  B )  =  ( A  |`  B )
4 rescnvcnv 5285 . 2  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
53, 4eqtr4i 2434 1  |-  `' `' ( A  |`  B )  =  ( `' `' A  |`  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405   `'ccnv 4821    |` cres 4824   Rel wrel 4827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-rel 4829  df-cnv 4830  df-res 4834
This theorem is referenced by: (None)
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