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Theorem ecqs 8721
Description: Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)
Hypothesis
Ref Expression
ecqs.1 𝑅 ∈ V
Assertion
Ref Expression
ecqs [𝐴]𝑅 = ({𝐴} / 𝑅)

Proof of Theorem ecqs
StepHypRef Expression
1 df-ec 8651 . 2 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 ecqs.1 . . 3 𝑅 ∈ V
3 uniqs 8717 . . 3 (𝑅 ∈ V → ({𝐴} / 𝑅) = (𝑅 “ {𝐴}))
42, 3ax-mp 5 . 2 ({𝐴} / 𝑅) = (𝑅 “ {𝐴})
51, 4eqtr4i 2768 1 [𝐴]𝑅 = ({𝐴} / 𝑅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  Vcvv 3446  {csn 4587   cuni 4866  cima 5637  [cec 8647   / cqs 8648
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-11 2155  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ec 8651  df-qs 8655
This theorem is referenced by: (None)
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