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Theorem cocnvcnv2 5888
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 5828 . . 3 𝐵 = (𝐵 ↾ V)
21coeq2i 5515 . 2 (𝐴𝐵) = (𝐴 ∘ (𝐵 ↾ V))
3 resco 5880 . 2 ((𝐴𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V))
4 relco 5874 . . 3 Rel (𝐴𝐵)
5 dfrel3 5832 . . 3 (Rel (𝐴𝐵) ↔ ((𝐴𝐵) ↾ V) = (𝐴𝐵))
64, 5mpbi 222 . 2 ((𝐴𝐵) ↾ V) = (𝐴𝐵)
72, 3, 63eqtr2i 2855 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1656  Vcvv 3414  ccnv 5341  cres 5344  ccom 5346  Rel wrel 5347
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 5005  ax-nul 5013  ax-pr 5127
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 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-res 5354
This theorem is referenced by:  dfdm2  5908  cofunex2g  7393  trclubgNEW  38759  cnvtrrel  38796  trrelsuperrel2dg  38797
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