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Theorem cnvcnvres 5008
Description: The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
Assertion
Ref Expression
cnvcnvres (𝐴𝐵) = (𝐴𝐵)

Proof of Theorem cnvcnvres
StepHypRef Expression
1 relres 4853 . . 3 Rel (𝐴𝐵)
2 dfrel2 4995 . . 3 (Rel (𝐴𝐵) ↔ (𝐴𝐵) = (𝐴𝐵))
31, 2mpbi 144 . 2 (𝐴𝐵) = (𝐴𝐵)
4 rescnvcnv 5007 . 2 (𝐴𝐵) = (𝐴𝐵)
53, 4eqtr4i 2164 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1332  ccnv 4544  cres 4547  Rel wrel 4550
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-br 3936  df-opab 3996  df-xp 4551  df-rel 4552  df-cnv 4553  df-res 4557
This theorem is referenced by: (None)
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