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	⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵)

Proof of Theorem cocnvcnv2

Step	Hyp	Ref	Expression
1		cnvcnv2 6189	. . 3 ⊢ ◡◡𝐵 = (𝐵 ↾ V)
2	1	coeq2i 5858	. 2 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ (𝐵 ↾ V))
3		resco 6246	. 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V))
4		relco 6104	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
5		dfrel3 6194	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵))
6	4, 5	mpbi 229	. 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵)
7	2, 3, 6	3eqtr2i 2766	1 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵)

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