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Theorem cocnvcnv2 5808
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 5746 . . 3 𝐵 = (𝐵 ↾ V)
21coeq2i 5438 . 2 (𝐴𝐵) = (𝐴 ∘ (𝐵 ↾ V))
3 resco 5800 . 2 ((𝐴𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V))
4 relco 5794 . . 3 Rel (𝐴𝐵)
5 dfrel3 5750 . . 3 (Rel (𝐴𝐵) ↔ ((𝐴𝐵) ↾ V) = (𝐴𝐵))
64, 5mpbi 220 . 2 ((𝐴𝐵) ↾ V) = (𝐴𝐵)
72, 3, 63eqtr2i 2788 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  Vcvv 3340  ccnv 5265  cres 5268  ccom 5270  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-res 5278
This theorem is referenced by:  dfdm2  5828  cofunex2g  7297  trclubgNEW  38445  cnvtrrel  38482  trrelsuperrel2dg  38483
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