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Theorem snec 6490
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 6464 . . . . 5 (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅)
32eqeq2d 2151 . . . 4 (𝑥 = 𝐴 → (𝑦 = [𝑥]𝑅𝑦 = [𝐴]𝑅))
41, 3rexsn 3568 . . 3 (∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅𝑦 = [𝐴]𝑅)
54abbii 2255 . 2 {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦𝑦 = [𝐴]𝑅}
6 df-qs 6435 . 2 ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅}
7 df-sn 3533 . 2 {[𝐴]𝑅} = {𝑦𝑦 = [𝐴]𝑅}
85, 6, 73eqtr4ri 2171 1 {[𝐴]𝑅} = ({𝐴} / 𝑅)
Colors of variables: wff set class
Syntax hints:   = wceq 1331  wcel 1480  {cab 2125  wrex 2417  Vcvv 2686  {csn 3527  [cec 6427   / cqs 6428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-ec 6431  df-qs 6435
This theorem is referenced by: (None)
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