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Theorem cocnvcnv2 6278
Description: A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
Assertion
Ref Expression
cocnvcnv2 (𝐴𝐵) = (𝐴𝐵)

Proof of Theorem cocnvcnv2
StepHypRef Expression
1 cnvcnv2 6213 . . 3 𝐵 = (𝐵 ↾ V)
21coeq2i 5871 . 2 (𝐴𝐵) = (𝐴 ∘ (𝐵 ↾ V))
3 resco 6270 . 2 ((𝐴𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V))
4 relco 6126 . . 3 Rel (𝐴𝐵)
5 dfrel3 6218 . . 3 (Rel (𝐴𝐵) ↔ ((𝐴𝐵) ↾ V) = (𝐴𝐵))
64, 5mpbi 230 . 2 ((𝐴𝐵) ↾ V) = (𝐴𝐵)
72, 3, 63eqtr2i 2771 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  Vcvv 3480  ccnv 5684  cres 5687  ccom 5689  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-res 5697
This theorem is referenced by:  dfdm2  6301  cofunex2g  7974  trclubgNEW  43631  cnvtrrel  43683  trrelsuperrel2dg  43684
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