Theorem cocnvcnv1 6276

Description: A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)

Assertion

Ref	Expression
cocnvcnv1	⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)

Proof of Theorem cocnvcnv1

Step	Hyp	Ref	Expression
1		cnvcnv2 6212	. . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2	1	coeq1i 5869	. 2 ⊢ (◡◡𝐴 ∘ 𝐵) = ((𝐴 ↾ V) ∘ 𝐵)
3		ssv 4007	. . 3 ⊢ ran 𝐵 ⊆ V
4		cores 6268	. . 3 ⊢ (ran 𝐵 ⊆ V → ((𝐴 ↾ V) ∘ 𝐵) = (𝐴 ∘ 𝐵))
5	3, 4	ax-mp 5	. 2 ⊢ ((𝐴 ↾ V) ∘ 𝐵) = (𝐴 ∘ 𝐵)
6	2, 5	eqtri 2764	1 ⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)

Syntax hints:   = wceq 1539  Vcvv 3479   ⊆ wss 3950  ^◡ccnv 5683  ran crn 5685   ↾ cres 5686   ∘ ccom 5688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696

This theorem is referenced by:  cores2  6278  coires1  6283  cofunex2g  7975  mvdco  19464  deg1val  26136  trlcocnv  40723  trclubgNEW  43636  cnvtrrel  43688  trrelsuperrel2dg  43689