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Theorem ecres2 36552
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 36550 . . . . 5 (𝑦 ∈ V → (𝑦 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝑦)))
21elv 3447 . . . 4 (𝑦 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝑦))
32baib 536 . . 3 (𝐵𝐴 → (𝑦 ∈ [𝐵](𝑅𝐴) ↔ 𝐵𝑅𝑦))
43abbi2dv 2875 . 2 (𝐵𝐴 → [𝐵](𝑅𝐴) = {𝑦𝐵𝑅𝑦})
5 dfec2 8572 . 2 (𝐵𝐴 → [𝐵]𝑅 = {𝑦𝐵𝑅𝑦})
64, 5eqtr4d 2779 1 (𝐵𝐴 → [𝐵](𝑅𝐴) = [𝐵]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  {cab 2713  Vcvv 3441   class class class wbr 5092  cres 5622  [cec 8567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-xp 5626  df-rel 5627  df-cnv 5628  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-ec 8571
This theorem is referenced by:  eccnvepres2  36558  eldmqsres  36560  qsresid  36599  ecex2  36602
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