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

Theorem elecres 38182
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 6034 . . 3 Rel (𝑅𝐴)
2 relelec 8806 . . 3 (Rel (𝑅𝐴) → (𝐶 ∈ [𝐵](𝑅𝐴) ↔ 𝐵(𝑅𝐴)𝐶))
31, 2ax-mp 5 . 2 (𝐶 ∈ [𝐵](𝑅𝐴) ↔ 𝐵(𝑅𝐴)𝐶)
4 brres 6015 . 2 (𝐶𝑉 → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))
53, 4bitrid 283 1 (𝐶𝑉 → (𝐶 ∈ [𝐵](𝑅𝐴) ↔ (𝐵𝐴𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2103   class class class wbr 5169  cres 5701  Rel wrel 5704  [cec 8757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-xp 5705  df-rel 5706  df-cnv 5707  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-ec 8761
This theorem is referenced by:  ecres  38183  ecres2  38184
  Copyright terms: Public domain W3C validator