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Theorem cnvcnvres 5562
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 5390 . . 3 Rel (𝐴𝐵)
2 dfrel2 5547 . . 3 (Rel (𝐴𝐵) ↔ (𝐴𝐵) = (𝐴𝐵))
31, 2mpbi 220 . 2 (𝐴𝐵) = (𝐴𝐵)
4 rescnvcnv 5561 . 2 (𝐴𝐵) = (𝐴𝐵)
53, 4eqtr4i 2646 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  ccnv 5078  cres 5081  Rel wrel 5084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-res 5091
This theorem is referenced by: (None)
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