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Theorem cnvcnvres 5136
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 4983 . . 3  |-  Rel  ( A  |`  B )
2 dfrel2 5124 . . 3  |-  ( Rel  ( A  |`  B )  <->  `' `' ( A  |`  B )  =  ( A  |`  B )
)
31, 2mpbi 199 . 2  |-  `' `' ( A  |`  B )  =  ( A  |`  B )
4 rescnvcnv 5135 . 2  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
53, 4eqtr4i 2306 1  |-  `' `' ( A  |`  B )  =  ( `' `' A  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   `'ccnv 4688    |` cres 4691   Rel wrel 4694
This theorem is referenced by:  domrancur1c  25202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-res 4701
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