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Mirrors > Home > MPE Home > Th. List > snec | Structured version Visualization version GIF version |
Description: The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
snec.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
snec | ⊢ {[𝐴]𝑅} = ({𝐴} / 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snec.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | eceq1 8321 | . . . . 5 ⊢ (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅) | |
3 | 2 | eqeq2d 2832 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅)) |
4 | 1, 3 | rexsn 4614 | . . 3 ⊢ (∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅) |
5 | 4 | abbii 2886 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅} |
6 | df-qs 8289 | . 2 ⊢ ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} | |
7 | df-sn 4562 | . 2 ⊢ {[𝐴]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅} | |
8 | 5, 6, 7 | 3eqtr4ri 2855 | 1 ⊢ {[𝐴]𝑅} = ({𝐴} / 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 {cab 2799 ∃wrex 3139 Vcvv 3495 {csn 4561 [cec 8281 / cqs 8282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 df-cnv 5558 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-ec 8285 df-qs 8289 |
This theorem is referenced by: (None) |
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