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Theorem snec 8720
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 8687 . . . . 5 (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅)
32eqeq2d 2748 . . . 4 (𝑥 = 𝐴 → (𝑦 = [𝑥]𝑅𝑦 = [𝐴]𝑅))
41, 3rexsn 4644 . . 3 (∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅𝑦 = [𝐴]𝑅)
54abbii 2807 . 2 {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦𝑦 = [𝐴]𝑅}
6 df-qs 8655 . 2 ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅}
7 df-sn 4588 . 2 {[𝐴]𝑅} = {𝑦𝑦 = [𝐴]𝑅}
85, 6, 73eqtr4ri 2776 1 {[𝐴]𝑅} = ({𝐴} / 𝑅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  {cab 2714  wrex 3074  Vcvv 3446  {csn 4587  [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-ext 2708
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-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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