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Theorem snec 6233
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 6207 . . . . 5 (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅)
32eqeq2d 2093 . . . 4 (𝑥 = 𝐴 → (𝑦 = [𝑥]𝑅𝑦 = [𝐴]𝑅))
41, 3rexsn 3445 . . 3 (∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅𝑦 = [𝐴]𝑅)
54abbii 2195 . 2 {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦𝑦 = [𝐴]𝑅}
6 df-qs 6178 . 2 ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅}
7 df-sn 3412 . 2 {[𝐴]𝑅} = {𝑦𝑦 = [𝐴]𝑅}
85, 6, 73eqtr4ri 2113 1 {[𝐴]𝑅} = ({𝐴} / 𝑅)
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wcel 1434  {cab 2068  wrex 2350  Vcvv 2602  {csn 3406  [cec 6170   / cqs 6171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-ec 6174  df-qs 6178
This theorem is referenced by: (None)
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