Theorem cocnvcnv1 6217

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 6152	. . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2	1	coeq1i 5809	. 2 ⊢ (◡◡𝐴 ∘ 𝐵) = ((𝐴 ↾ V) ∘ 𝐵)
3		ssv 3959	. . 3 ⊢ ran 𝐵 ⊆ V
4		cores 6208	. . 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 2760	1 ⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)

Syntax hints:   = wceq 1542  Vcvv 3441   ⊆ wss 3902  ^◡ccnv 5624  ran crn 5626   ↾ cres 5627   ∘ ccom 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637

This theorem is referenced by:  cores2  6219  coires1  6224  cofunex2g  7897  mvdco  19379  deg1val  26062  trlcocnv  41059  trclubgNEW  43937  cnvtrrel  43989  trrelsuperrel2dg  43990