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Theorem ecres2 38261
Description: The restricted coset of 𝐵 when 𝐵 is an element of the restriction. (Contributed by Peter Mazsa, 16-Oct-2018.)
Assertion
Ref Expression
ecres2 (𝐵𝐴 → [𝐵](𝑅𝐴) = [𝐵]𝑅)

Proof of Theorem ecres2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elecres 38259 . . . . 5 (𝑦 ∈ V → (𝑦 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝑦)))
21elv 3483 . . . 4 (𝑦 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝑦))
32baib 535 . . 3 (𝐵𝐴 → (𝑦 ∈ [𝐵](𝑅𝐴) ↔ 𝐵𝑅𝑦))
43eqabdv 2873 . 2 (𝐵𝐴 → [𝐵](𝑅𝐴) = {𝑦𝐵𝑅𝑦})
5 dfec2 8747 . 2 (𝐵𝐴 → [𝐵]𝑅 = {𝑦𝐵𝑅𝑦})
64, 5eqtr4d 2778 1 (𝐵𝐴 → [𝐵](𝑅𝐴) = [𝐵]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  Vcvv 3478   class class class wbr 5148  cres 5691  [cec 8742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746
This theorem is referenced by:  eccnvepres2  38267  eldmqsres  38269  qsresid  38307  ecex2  38310
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