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Theorem cnvcnv2 6204
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 6203 . 2 𝐴 = (𝐴 ∩ (V × V))
2 df-res 5694 . 2 (𝐴 ↾ V) = (𝐴 ∩ (V × V))
31, 2eqtr4i 2757 1 𝐴 = (𝐴 ↾ V)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  Vcvv 3462  cin 3946   × cxp 5680  ccnv 5681  cres 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689  df-cnv 5690  df-res 5694
This theorem is referenced by:  dfrel3  6209  rnresv  6212  rescnvcnv  6215  cocnvcnv1  6268  cocnvcnv2  6269  strfv2d  17204  resnonrel  43259
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