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Theorem rescnvcnv 6224
Description: The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
rescnvcnv (𝐴𝐵) = (𝐴𝐵)

Proof of Theorem rescnvcnv
StepHypRef Expression
1 cnvcnv2 6213 . . 3 𝐴 = (𝐴 ↾ V)
21reseq1i 5993 . 2 (𝐴𝐵) = ((𝐴 ↾ V) ↾ 𝐵)
3 resres 6010 . 2 ((𝐴 ↾ V) ↾ 𝐵) = (𝐴 ↾ (V ∩ 𝐵))
4 ssv 4008 . . . 4 𝐵 ⊆ V
5 sseqin2 4223 . . . 4 (𝐵 ⊆ V ↔ (V ∩ 𝐵) = 𝐵)
64, 5mpbi 230 . . 3 (V ∩ 𝐵) = 𝐵
76reseq2i 5994 . 2 (𝐴 ↾ (V ∩ 𝐵)) = (𝐴𝐵)
82, 3, 73eqtri 2769 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  Vcvv 3480  cin 3950  wss 3951  ccnv 5684  cres 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-res 5697
This theorem is referenced by:  cnvcnvres  6225  imacnvcnv  6226  resdm2  6251  resdmres  6252  coires1  6284  f1oresrab  7147
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