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Theorem elecres 37753
Description: Elementhood in the restricted coset of 𝐵. (Contributed by Peter Mazsa, 21-Sep-2018.)
Assertion
Ref Expression
elecres (𝐶𝑉 → (𝐶 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝐶)))

Proof of Theorem elecres
StepHypRef Expression
1 relres 6013 . . 3 Rel (𝑅𝐴)
2 relelec 8775 . . 3 (Rel (𝑅𝐴) → (𝐶 ∈ [𝐵](𝑅𝐴) ↔ 𝐵(𝑅𝐴)𝐶))
31, 2ax-mp 5 . 2 (𝐶 ∈ [𝐵](𝑅𝐴) ↔ 𝐵(𝑅𝐴)𝐶)
4 brres 5994 . 2 (𝐶𝑉 → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))
53, 4bitrid 282 1 (𝐶𝑉 → (𝐶 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098   class class class wbr 5150  cres 5682  Rel wrel 5685  [cec 8727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-xp 5686  df-rel 5687  df-cnv 5688  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-ec 8731
This theorem is referenced by:  ecres  37754  ecres2  37755
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