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Theorem ecres 34540
 Description: Restricted coset of 𝐵. (Contributed by Peter Mazsa, 9-Dec-2018.)
Assertion
Ref Expression
ecres [𝐵](𝑅𝐴) = {𝑥 ∣ (𝐵𝐴𝐵𝑅𝑥)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem ecres
StepHypRef Expression
1 elecres 34539 . . 3 (𝑥 ∈ V → (𝑥 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝑥)))
21elv 3389 . 2 (𝑥 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝑥))
32abbi2i 2915 1 [𝐵](𝑅𝐴) = {𝑥 ∣ (𝐵𝐴𝐵𝑅𝑥)}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157  {cab 2785  Vcvv 3385   class class class wbr 4843   ↾ cres 5314  [cec 7980 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-ec 7984 This theorem is referenced by:  eccnvepres  34542
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