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Theorem snec 6066
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 6040 . . . . 5 (x = A → [x]𝑅 = [A]𝑅)
32eqeq2d 2024 . . . 4 (x = A → (y = [x]𝑅y = [A]𝑅))
41, 3rexsn 3378 . . 3 (x {A}y = [x]𝑅y = [A]𝑅)
54abbii 2126 . 2 {yx {A}y = [x]𝑅} = {yy = [A]𝑅}
6 df-qs 6011 . 2 ({A} / 𝑅) = {yx {A}y = [x]𝑅}
7 df-sn 3345 . 2 {[A]𝑅} = {yy = [A]𝑅}
85, 6, 73eqtr4ri 2044 1 {[A]𝑅} = ({A} / 𝑅)
Colors of variables: wff set class
Syntax hints:   = wceq 1223   wcel 1366  {cab 1999  wrex 2276  Vcvv 2526  {csn 3339  [cec 6003   / cqs 6004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-rex 2281  df-v 2528  df-sbc 2733  df-un 2890  df-in 2892  df-ss 2899  df-sn 3345  df-pr 3346  df-op 3348  df-br 3728  df-opab 3782  df-xp 4266  df-cnv 4268  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-ec 6007  df-qs 6011
This theorem is referenced by: (None)
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