Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elecres Structured version   Visualization version   GIF version

Theorem elecres 34366
 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 5584 . . 3 Rel (𝑅𝐴)
2 relelec 7954 . . 3 (Rel (𝑅𝐴) → (𝐶 ∈ [𝐵](𝑅𝐴) ↔ 𝐵(𝑅𝐴)𝐶))
31, 2ax-mp 5 . 2 (𝐶 ∈ [𝐵](𝑅𝐴) ↔ 𝐵(𝑅𝐴)𝐶)
4 brresALTV 34356 . 2 (𝐶𝑉 → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))
53, 4syl5bb 272 1 (𝐶𝑉 → (𝐶 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∈ wcel 2139   class class class wbr 4804   ↾ cres 5268  Rel wrel 5271  [cec 7909 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ec 7913 This theorem is referenced by:  ecres  34367  ecres2  34368
 Copyright terms: Public domain W3C validator