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Theorem cnvcnvres 5403
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 5239 . . 3  |-  Rel  ( A  |`  B )
2 dfrel2 5389 . . 3  |-  ( Rel  ( A  |`  B )  <->  `' `' ( A  |`  B )  =  ( A  |`  B )
)
31, 2mpbi 208 . 2  |-  `' `' ( A  |`  B )  =  ( A  |`  B )
4 rescnvcnv 5402 . 2  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
53, 4eqtr4i 2483 1  |-  `' `' ( A  |`  B )  =  ( `' `' A  |`  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   `'ccnv 4940    |` cres 4943   Rel wrel 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-cnv 4949  df-res 4953
This theorem is referenced by: (None)
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