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Theorem ecres2 34366
 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 34364 . . . . 5 (𝑦 ∈ V → (𝑦 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝑦)))
21elv 34307 . . . 4 (𝑦 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝑦))
32baib 982 . . 3 (𝐵𝐴 → (𝑦 ∈ [𝐵](𝑅𝐴) ↔ 𝐵𝑅𝑦))
43abbi2dv 2878 . 2 (𝐵𝐴 → [𝐵](𝑅𝐴) = {𝑦𝐵𝑅𝑦})
5 dfec2 7912 . 2 (𝐵𝐴 → [𝐵]𝑅 = {𝑦𝐵𝑅𝑦})
64, 5eqtr4d 2795 1 (𝐵𝐴 → [𝐵](𝑅𝐴) = [𝐵]𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1630   ∈ wcel 2137  {cab 2744  Vcvv 3338   class class class wbr 4802   ↾ cres 5266  [cec 7907 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863  df-xp 5270  df-rel 5271  df-cnv 5272  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-ec 7911 This theorem is referenced by:  eccnvepres2  34371  eldmqsres  34373  qsresid  34418  ecex2  34422
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