Theorem snecg 8761

Description: The singleton of a coset is the singleton quotient. (Contributed by Peter Mazsa, 25-Mar-2019.)

Assertion

Ref	Expression
snecg	⊢ (𝐴 ∈ 𝑉 → {[𝐴]𝑅} = ({𝐴} / 𝑅))

Proof of Theorem snecg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eceq1 8720	. . . . . 6 ⊢ (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅)
2	1	eqeq2d 2775	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅))
3	2	rexsng 4637	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅))
4	3	abbidv 2830	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅})
5		df-qs 8686	. . 3 ⊢ ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅}
6		df-sn 4585	. . 3 ⊢ {[𝐴]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
7	4, 5, 6	3eqtr4g 2824	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} / 𝑅) = {[𝐴]𝑅})
8	7	eqcomd 2770	1 ⊢ (𝐴 ∈ 𝑉 → {[𝐴]𝑅} = ({𝐴} / 𝑅))

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144  {cab 2742  ∃wrex 3088  {csn 4584  [cec 8678   / cqs 8679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ec 8682  df-qs 8686

This theorem is referenced by:  ecqmap2  38954