Theorem cocnvcnv2 6259

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

Assertion

Ref	Expression
cocnvcnv2	⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵)

Proof of Theorem cocnvcnv2

Step	Hyp	Ref	Expression
1		cnvcnv2 6190	. . 3 ⊢ ◡◡𝐵 = (𝐵 ↾ V)
2	1	coeq2i 5844	. 2 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ (𝐵 ↾ V))
3		resco 6250	. 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V))
4		relco 6108	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
5		dfrel3 6196	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵))
6	4, 5	mpbi 233	. 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵)
7	2, 3, 6	3eqtr2i 2791	1 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵)

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3453  ^◡ccnv 5658   ↾ cres 5661   ∘ ccom 5663  Rel wrel 5664

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 2734  ax-sep 5255  ax-pr 5402

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-res 5671

This theorem is used by:  relcnvtrg  6267  dfdm2  6283  cofunex2g  7950  trclubgNEW  44445  cnvtrrel  44497  trrelsuperrel2dg  44498