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Theorem ecres2 37977
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 37975 . . . . 5 (𝑦 ∈ V → (𝑦 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝑦)))
21elv 3468 . . . 4 (𝑦 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝑦))
32baib 534 . . 3 (𝐵𝐴 → (𝑦 ∈ [𝐵](𝑅𝐴) ↔ 𝐵𝑅𝑦))
43eqabdv 2860 . 2 (𝐵𝐴 → [𝐵](𝑅𝐴) = {𝑦𝐵𝑅𝑦})
5 dfec2 8737 . 2 (𝐵𝐴 → [𝐵]𝑅 = {𝑦𝐵𝑅𝑦})
64, 5eqtr4d 2769 1 (𝐵𝐴 → [𝐵](𝑅𝐴) = [𝐵]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  {cab 2703  Vcvv 3462   class class class wbr 5153  cres 5684  [cec 8732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ec 8736
This theorem is referenced by:  eccnvepres2  37983  eldmqsres  37985  qsresid  38023  ecex2  38026
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