Theorem snecg 8823

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 8782	. . . . . 6 ⊢ (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅)
2	1	eqeq2d 2774	. . . . 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 2829	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅})
5		df-qs 8748	. . 3 ⊢ ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅}
6		df-sn 4589	. . 3 ⊢ {[𝐴]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
7	4, 5, 6	3eqtr4g 2823	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} / 𝑅) = {[𝐴]𝑅})
8	7	eqcomd 2769	1 ⊢ (𝐴 ∈ 𝑉 → {[𝐴]𝑅} = ({𝐴} / 𝑅))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090  {csn 4588  [cec 8740   / cqs 8741

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8744  df-qs 8748

This theorem is used by:  ecqmap2  39573