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Theorem cocnvcnv1 6250
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 6186 . . 3 𝐴 = (𝐴 ↾ V)
21coeq1i 5853 . 2 (𝐴𝐵) = ((𝐴 ↾ V) ∘ 𝐵)
3 ssv 4001 . . 3 ran 𝐵 ⊆ V
4 cores 6242 . . 3 (ran 𝐵 ⊆ V → ((𝐴 ↾ V) ∘ 𝐵) = (𝐴𝐵))
53, 4ax-mp 5 . 2 ((𝐴 ↾ V) ∘ 𝐵) = (𝐴𝐵)
62, 5eqtri 2754 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  Vcvv 3468  wss 3943  ccnv 5668  ran crn 5670  cres 5671  ccom 5673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681
This theorem is referenced by:  cores2  6252  coires1  6257  cofunex2g  7935  mvdco  19365  deg1val  25987  trlcocnv  40104  trclubgNEW  42945  cnvtrrel  42997  trrelsuperrel2dg  42998
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