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Theorem cocnvcnv1 6210
Description: A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
Assertion
Ref Expression
cocnvcnv1 (𝐴𝐵) = (𝐴𝐵)

Proof of Theorem cocnvcnv1
StepHypRef Expression
1 cnvcnv2 6146 . . 3 𝐴 = (𝐴 ↾ V)
21coeq1i 5816 . 2 (𝐴𝐵) = ((𝐴 ↾ V) ∘ 𝐵)
3 ssv 3969 . . 3 ran 𝐵 ⊆ V
4 cores 6202 . . 3 (ran 𝐵 ⊆ V → ((𝐴 ↾ V) ∘ 𝐵) = (𝐴𝐵))
53, 4ax-mp 5 . 2 ((𝐴 ↾ V) ∘ 𝐵) = (𝐴𝐵)
62, 5eqtri 2761 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  Vcvv 3444  wss 3911  ccnv 5633  ran crn 5635  cres 5636  ccom 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646
This theorem is referenced by:  cores2  6212  coires1  6217  cofunex2g  7883  mvdco  19232  deg1val  25477  trlcocnv  39229  trclubgNEW  41978  cnvtrrel  42030  trrelsuperrel2dg  42031
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