MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cocnvcnv2 Structured version   Visualization version   GIF version

Theorem cocnvcnv2 6254
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 6189 . . 3 𝐵 = (𝐵 ↾ V)
21coeq2i 5858 . 2 (𝐴𝐵) = (𝐴 ∘ (𝐵 ↾ V))
3 resco 6246 . 2 ((𝐴𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V))
4 relco 6104 . . 3 Rel (𝐴𝐵)
5 dfrel3 6194 . . 3 (Rel (𝐴𝐵) ↔ ((𝐴𝐵) ↾ V) = (𝐴𝐵))
64, 5mpbi 229 . 2 ((𝐴𝐵) ↾ V) = (𝐴𝐵)
72, 3, 63eqtr2i 2766 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  Vcvv 3474  ccnv 5674  cres 5677  ccom 5679  Rel wrel 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-res 5687
This theorem is referenced by:  dfdm2  6277  cofunex2g  7932  trclubgNEW  42354  cnvtrrel  42406  trrelsuperrel2dg  42407
  Copyright terms: Public domain W3C validator