Theorem snecg 8718

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 8677	. . . . . 6 ⊢ (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅)
2	1	eqeq2d 2748	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅))
3	2	rexsng 4634	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅))
4	3	abbidv 2803	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅})
5		df-qs 8643	. . 3 ⊢ ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅}
6		df-sn 4582	. . 3 ⊢ {[𝐴]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
7	4, 5, 6	3eqtr4g 2797	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} / 𝑅) = {[𝐴]𝑅})
8	7	eqcomd 2743	1 ⊢ (𝐴 ∈ 𝑉 → {[𝐴]𝑅} = ({𝐴} / 𝑅))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3061  {csn 4581  [cec 8635   / cqs 8636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639  df-qs 8643

This theorem is referenced by:  ecqmap2  38653