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Theorem ecqs 8820
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 8746 . 2 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 ecqs.1 . . 3 𝑅 ∈ V
3 uniqs 8816 . . 3 (𝑅 ∈ V → ({𝐴} / 𝑅) = (𝑅 “ {𝐴}))
42, 3ax-mp 5 . 2 ({𝐴} / 𝑅) = (𝑅 “ {𝐴})
51, 4eqtr4i 2766 1 [𝐴]𝑅 = ({𝐴} / 𝑅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631   cuni 4912  cima 5692  [cec 8742   / cqs 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-qs 8750
This theorem is referenced by: (None)
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