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Theorem rescnvcnv 6042
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 6031 . . 3 𝐴 = (𝐴 ↾ V)
21reseq1i 5830 . 2 (𝐴𝐵) = ((𝐴 ↾ V) ↾ 𝐵)
3 resres 5847 . 2 ((𝐴 ↾ V) ↾ 𝐵) = (𝐴 ↾ (V ∩ 𝐵))
4 ssv 3975 . . . 4 𝐵 ⊆ V
5 sseqin2 4175 . . . 4 (𝐵 ⊆ V ↔ (V ∩ 𝐵) = 𝐵)
64, 5mpbi 233 . . 3 (V ∩ 𝐵) = 𝐵
76reseq2i 5831 . 2 (𝐴 ↾ (V ∩ 𝐵)) = (𝐴𝐵)
82, 3, 73eqtri 2851 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  Vcvv 3479  cin 3917  wss 3918  ccnv 5535  cres 5538
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5048  df-opab 5110  df-xp 5542  df-rel 5543  df-cnv 5544  df-res 5548
This theorem is referenced by:  cnvcnvres  6043  imacnvcnv  6044  resdm2  6069  resdmres  6070  coires1  6098  f1oresrab  6870
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