Theorem cocnvcnv1 6248

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 6180	. . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2	1	coeq1i 5833	. 2 ⊢ (◡◡𝐴 ∘ 𝐵) = ((𝐴 ↾ V) ∘ 𝐵)
3		ssv 3954	. . 3 ⊢ ran 𝐵 ⊆ V
4		cores 6239	. . 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 2783	1 ⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3450   ⊆ wss 3898  ^◡ccnv 5646  ran crn 5648   ↾ cres 5649   ∘ ccom 5651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659

This theorem is used by:  cores2  6250  coires1  6255  relcnvtrg  6257  cofunex2g  7945  mvdco  19621  deg1val  26376  trlcocnv  41697  trclubgNEW  44562  cnvtrrel  44614  trrelsuperrel2dg  44615