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Theorem dfqs2 40972
Description: Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
Assertion
Ref Expression
dfqs2 (𝐴 / 𝑅) = ran (𝑥𝐴 ↦ [𝑥]𝑅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem dfqs2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-qs 8697 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
2 eqid 2733 . . 3 (𝑥𝐴 ↦ [𝑥]𝑅) = (𝑥𝐴 ↦ [𝑥]𝑅)
32rnmpt 5949 . 2 ran (𝑥𝐴 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
41, 3eqtr4i 2764 1 (𝐴 / 𝑅) = ran (𝑥𝐴 ↦ [𝑥]𝑅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  {cab 2710  wrex 3071  cmpt 5227  ran crn 5673  [cec 8689   / cqs 8690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-mpt 5228  df-cnv 5680  df-dm 5682  df-rn 5683  df-qs 8697
This theorem is referenced by:  qsalrel  40975
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