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Theorem snec 8080
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 8052 . . . . 5 (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅)
32eqeq2d 2835 . . . 4 (𝑥 = 𝐴 → (𝑦 = [𝑥]𝑅𝑦 = [𝐴]𝑅))
41, 3rexsn 4445 . . 3 (∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅𝑦 = [𝐴]𝑅)
54abbii 2944 . 2 {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦𝑦 = [𝐴]𝑅}
6 df-qs 8020 . 2 ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅}
7 df-sn 4400 . 2 {[𝐴]𝑅} = {𝑦𝑦 = [𝐴]𝑅}
85, 6, 73eqtr4ri 2860 1 {[𝐴]𝑅} = ({𝐴} / 𝑅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1656  wcel 2164  {cab 2811  wrex 3118  Vcvv 3414  {csn 4399  [cec 8012   / cqs 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-cnv 5354  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-ec 8016  df-qs 8020
This theorem is referenced by: (None)
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