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Theorem cocnvcnv2 5611
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 5552 . . 3 𝐵 = (𝐵 ↾ V)
21coeq2i 5247 . 2 (𝐴𝐵) = (𝐴 ∘ (𝐵 ↾ V))
3 resco 5603 . 2 ((𝐴𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V))
4 relco 5597 . . 3 Rel (𝐴𝐵)
5 dfrel3 5556 . . 3 (Rel (𝐴𝐵) ↔ ((𝐴𝐵) ↾ V) = (𝐴𝐵))
64, 5mpbi 220 . 2 ((𝐴𝐵) ↾ V) = (𝐴𝐵)
72, 3, 63eqtr2i 2649 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  Vcvv 3189  ccnv 5078  cres 5081  ccom 5083  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-rex 2913  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-co 5088  df-res 5091
This theorem is referenced by:  dfdm2  5631  cofunex2g  7085  trclubgNEW  37441  cnvtrrel  37478  trrelsuperrel2dg  37479
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