Theorem snec 6190

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.

Step	Hyp	Ref	Expression
1		snec.1	. . . 4 ⊢ 𝐴 ∈ V
2		eceq1 6164	. . . . 5 ⊢ (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅)
3	2	eqeq2d 2092	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅))
4	1, 3	rexsn 3437	. . 3 ⊢ (∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅)
5	4	abbii 2194	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
6		df-qs 6135	. 2 ⊢ ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅}
7		df-sn 3404	. 2 ⊢ {[𝐴]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
8	5, 6, 7	3eqtr4ri 2112	1 ⊢ {[𝐴]𝑅} = ({𝐴} / 𝑅)

Syntax hints:   = wceq 1284   ∈ wcel 1433  {cab 2067  ∃wrex 2349  Vcvv 2601  {csn 3398  [cec 6127   / cqs 6128

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-ec 6131  df-qs 6135

This theorem is referenced by: (None)