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Theorem ecqs 8772
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 8702 . 2 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 ecqs.1 . . 3 𝑅 ∈ V
3 uniqs 8768 . . 3 (𝑅 ∈ V → ({𝐴} / 𝑅) = (𝑅 “ {𝐴}))
42, 3ax-mp 5 . 2 ({𝐴} / 𝑅) = (𝑅 “ {𝐴})
51, 4eqtr4i 2755 1 [𝐴]𝑅 = ({𝐴} / 𝑅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  Vcvv 3466  {csn 4621   cuni 4900  cima 5670  [cec 8698   / cqs 8699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-xp 5673  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ec 8702  df-qs 8706
This theorem is referenced by: (None)
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