Theorem cocnvcnv2 6106

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 6045	. . 3 ⊢ ◡◡𝐵 = (𝐵 ↾ V)
2	1	coeq2i 5726	. 2 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ (𝐵 ↾ V))
3		resco 6098	. 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V))
4		relco 6092	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
5		dfrel3 6050	. . 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 2850	1 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵)

Syntax hints:   = wceq 1533  Vcvv 3495  ^◡ccnv 5549   ↾ cres 5552   ∘ ccom 5554  Rel wrel 5555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-res 5562

This theorem is referenced by:  dfdm2  6127  cofunex2g  7645  trclubgNEW  39971  cnvtrrel  40008  trrelsuperrel2dg  40009