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Theorem cocnvcnv1 5605
 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 5547 . . 3 𝐴 = (𝐴 ↾ V)
21coeq1i 5241 . 2 (𝐴𝐵) = ((𝐴 ↾ V) ∘ 𝐵)
3 ssv 3604 . . 3 ran 𝐵 ⊆ V
4 cores 5597 . . 3 (ran 𝐵 ⊆ V → ((𝐴 ↾ V) ∘ 𝐵) = (𝐴𝐵))
53, 4ax-mp 5 . 2 ((𝐴 ↾ V) ∘ 𝐵) = (𝐴𝐵)
62, 5eqtri 2643 1 (𝐴𝐵) = (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480  Vcvv 3186   ⊆ wss 3555  ◡ccnv 5073  ran crn 5075   ↾ cres 5076   ∘ ccom 5078 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 4741  ax-nul 4749  ax-pr 4867 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 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086 This theorem is referenced by:  cores2  5607  coires1  5612  cofunex2g  7078  mvdco  17786  deg1val  23760  trlcocnv  35488  trclubgNEW  37406  cnvtrrel  37443  trrelsuperrel2dg  37444
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