Theorem snecg 8780

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 8739	. . . . . 6 ⊢ (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅)
2	1	eqeq2d 2773	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅))
3	2	rexsng 4640	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅))
4	3	abbidv 2828	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅})
5		df-qs 8705	. . 3 ⊢ ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅}
6		df-sn 4588	. . 3 ⊢ {[𝐴]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
7	4, 5, 6	3eqtr4g 2822	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} / 𝑅) = {[𝐴]𝑅})
8	7	eqcomd 2768	1 ⊢ (𝐴 ∈ 𝑉 → {[𝐴]𝑅} = ({𝐴} / 𝑅))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {cab 2740  ∃wrex 3088  {csn 4587  [cec 8697   / cqs 8698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8701  df-qs 8705

This theorem is used by:  ecqmap2  39185