Theorem cocnvcnv1 6246

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 6182	. . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2	1	coeq1i 5849	. 2 ⊢ (◡◡𝐴 ∘ 𝐵) = ((𝐴 ↾ V) ∘ 𝐵)
3		ssv 3998	. . 3 ⊢ ran 𝐵 ⊆ V
4		cores 6238	. . 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 2752	1 ⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)

Syntax hints:   = wceq 1533  Vcvv 3466   ⊆ wss 3940  ^◡ccnv 5665  ran crn 5667   ↾ cres 5668   ∘ ccom 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678

This theorem is referenced by:  cores2  6248  coires1  6253  cofunex2g  7929  mvdco  19355  deg1val  25954  trlcocnv  40081  trclubgNEW  42858  cnvtrrel  42910  trrelsuperrel2dg  42911