Theorem cocnvcnv1 6258

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 6190	. . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2	1	coeq1i 5844	. 2 ⊢ (◡◡𝐴 ∘ 𝐵) = ((𝐴 ↾ V) ∘ 𝐵)
3		ssv 3960	. . 3 ⊢ ran 𝐵 ⊆ V
4		cores 6249	. . 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 2785	1 ⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3454   ⊆ wss 3904  ^◡ccnv 5659  ran crn 5661   ↾ cres 5662   ∘ ccom 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672

This theorem is used by:  cores2  6260  coires1  6265  relcnvtrg  6267  cofunex2g  7945  mvdco  19521  deg1val  26264  trlcocnv  41522  trclubgNEW  44372  cnvtrrel  44424  trrelsuperrel2dg  44425