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Theorem ecqs 7967
 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 7902 . 2 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 ecqs.1 . . 3 𝑅 ∈ V
3 uniqs 7963 . . 3 (𝑅 ∈ V → ({𝐴} / 𝑅) = (𝑅 “ {𝐴}))
42, 3ax-mp 5 . 2 ({𝐴} / 𝑅) = (𝑅 “ {𝐴})
51, 4eqtr4i 2796 1 [𝐴]𝑅 = ({𝐴} / 𝑅)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631   ∈ wcel 2145  Vcvv 3351  {csn 4317  ∪ cuni 4575   “ cima 5253  [cec 7898   / cqs 7899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ec 7902  df-qs 7906 This theorem is referenced by: (None)
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