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Theorem cocnvcnv1 5891
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 5832 . . 3 𝐴 = (𝐴 ↾ V)
21coeq1i 5518 . 2 (𝐴𝐵) = ((𝐴 ↾ V) ∘ 𝐵)
3 ssv 3850 . . 3 ran 𝐵 ⊆ V
4 cores 5883 . . 3 (ran 𝐵 ⊆ V → ((𝐴 ↾ V) ∘ 𝐵) = (𝐴𝐵))
53, 4ax-mp 5 . 2 ((𝐴 ↾ V) ∘ 𝐵) = (𝐴𝐵)
62, 5eqtri 2849 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1656  Vcvv 3414  wss 3798  ccnv 5345  ran crn 5347  cres 5348  ccom 5350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358
This theorem is referenced by:  cores2  5893  coires1  5898  cofunex2g  7398  mvdco  18222  deg1val  24262  trlcocnv  36790  trclubgNEW  38761  cnvtrrel  38798  trrelsuperrel2dg  38799
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