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

Theorem qsresid 34590
Description: Simplification of a special quotient set. (Contributed by Peter Mazsa, 2-Sep-2020.)
Assertion
Ref Expression
qsresid (𝐴 / (𝑅𝐴)) = (𝐴 / 𝑅)

Proof of Theorem qsresid
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecres2 34541 . . . . 5 (𝑣𝐴 → [𝑣](𝑅𝐴) = [𝑣]𝑅)
21eqeq2d 2809 . . . 4 (𝑣𝐴 → (𝑢 = [𝑣](𝑅𝐴) ↔ 𝑢 = [𝑣]𝑅))
32rexbiia 3221 . . 3 (∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴) ↔ ∃𝑣𝐴 𝑢 = [𝑣]𝑅)
43abbii 2916 . 2 {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴)} = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣]𝑅}
5 df-qs 7988 . 2 (𝐴 / (𝑅𝐴)) = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴)}
6 df-qs 7988 . 2 (𝐴 / 𝑅) = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣]𝑅}
74, 5, 63eqtr4i 2831 1 (𝐴 / (𝑅𝐴)) = (𝐴 / 𝑅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wcel 2157  {cab 2785  wrex 3090  cres 5314  [cec 7980   / cqs 7981
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  df-qs 7988
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator