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Theorem snec 8792
Description: The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypothesis
Ref Expression
snec.1 𝐴 ∈ V
Assertion
Ref Expression
snec {[𝐴]𝑅} = ({𝐴} / 𝑅)

Proof of Theorem snec
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snec.1 . . . 4 𝐴 ∈ V
2 eceq1 8756 . . . . 5 (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅)
32eqeq2d 2739 . . . 4 (𝑥 = 𝐴 → (𝑦 = [𝑥]𝑅𝑦 = [𝐴]𝑅))
41, 3rexsn 4682 . . 3 (∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅𝑦 = [𝐴]𝑅)
54abbii 2798 . 2 {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦𝑦 = [𝐴]𝑅}
6 df-qs 8724 . 2 ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅}
7 df-sn 4625 . 2 {[𝐴]𝑅} = {𝑦𝑦 = [𝐴]𝑅}
85, 6, 73eqtr4ri 2767 1 {[𝐴]𝑅} = ({𝐴} / 𝑅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  {cab 2705  wrex 3066  Vcvv 3470  {csn 4624  [cec 8716   / cqs 8717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8720  df-qs 8724
This theorem is referenced by: (None)
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