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Theorem ecres2 34594
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 34592 . . . . 5 (𝑦 ∈ V → (𝑦 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝑦)))
21elv 3418 . . . 4 (𝑦 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝑦))
32baib 531 . . 3 (𝐵𝐴 → (𝑦 ∈ [𝐵](𝑅𝐴) ↔ 𝐵𝑅𝑦))
43abbi2dv 2947 . 2 (𝐵𝐴 → [𝐵](𝑅𝐴) = {𝑦𝐵𝑅𝑦})
5 dfec2 8017 . 2 (𝐵𝐴 → [𝐵]𝑅 = {𝑦𝐵𝑅𝑦})
64, 5eqtr4d 2864 1 (𝐵𝐴 → [𝐵](𝑅𝐴) = [𝐵]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164  {cab 2811  Vcvv 3414   class class class wbr 4875  cres 5348  [cec 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-rel 5353  df-cnv 5354  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-ec 8016
This theorem is referenced by:  eccnvepres2  34599  eldmqsres  34601  qsresid  34643  ecex2  34647
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