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Theorem cnvcnv2 6215
Description: The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)
Assertion
Ref Expression
cnvcnv2 𝐴 = (𝐴 ↾ V)

Proof of Theorem cnvcnv2
StepHypRef Expression
1 cnvcnv 6214 . 2 𝐴 = (𝐴 ∩ (V × V))
2 df-res 5701 . 2 (𝐴 ↾ V) = (𝐴 ∩ (V × V))
31, 2eqtr4i 2766 1 𝐴 = (𝐴 ↾ V)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3478  cin 3962   × cxp 5687  ccnv 5688  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-res 5701
This theorem is referenced by:  dfrel3  6220  rnresv  6223  rescnvcnv  6226  cocnvcnv1  6279  cocnvcnv2  6280  strfv2d  17236  resnonrel  43582
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