Theorem cocnvcnv2 5888

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 5828	. . 3 ⊢ ◡◡𝐵 = (𝐵 ↾ V)
2	1	coeq2i 5515	. 2 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ (𝐵 ↾ V))
3		resco 5880	. 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V))
4		relco 5874	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
5		dfrel3 5832	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵))
6	4, 5	mpbi 222	. 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵)
7	2, 3, 6	3eqtr2i 2855	1 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵)

Syntax hints:   = wceq 1656  Vcvv 3414  ^◡ccnv 5341   ↾ cres 5344   ∘ ccom 5346  Rel wrel 5347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-res 5354

This theorem is referenced by:  dfdm2  5908  cofunex2g  7393  trclubgNEW  38759  cnvtrrel  38796  trrelsuperrel2dg  38797