Theorem cocnvcnv2 6248

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 6181	. . 3 ⊢ ◡◡𝐵 = (𝐵 ↾ V)
2	1	coeq2i 5834	. 2 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ (𝐵 ↾ V))
3		resco 6239	. 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V))
4		relco 6099	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
5		dfrel3 6187	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵))
6	4, 5	mpbi 232	. 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵)
7	2, 3, 6	3eqtr2i 2793	1 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵)

Syntax hints:   = wceq 1562  Vcvv 3456  ^◡ccnv 5648   ↾ cres 5651   ∘ ccom 5653  Rel wrel 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-res 5661

This theorem is referenced by:  dfdm2  6270  cofunex2g  7933  trclubgNEW  44199  cnvtrrel  44251  trrelsuperrel2dg  44252