Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfqs3 Structured version   Visualization version   GIF version

Theorem dfqs3 42279
Description: Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
Assertion
Ref Expression
dfqs3 (𝐴 / 𝑅) = 𝑥𝐴 {[𝑥]𝑅}
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem dfqs3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-qs 8751 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
2 iunsn 5066 . 2 𝑥𝐴 {[𝑥]𝑅} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
31, 2eqtr4i 2768 1 (𝐴 / 𝑅) = 𝑥𝐴 {[𝑥]𝑅}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  {cab 2714  wrex 3070  {csn 4626   ciun 4991  [cec 8743   / cqs 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-v 3482  df-sn 4627  df-iun 4993  df-qs 8751
This theorem is referenced by:  prjspval2  42623
  Copyright terms: Public domain W3C validator