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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 8756 | . . . . 5 ⊢ (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅) | |
3 | 2 | eqeq2d 2739 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅)) |
4 | 1, 3 | rexsn 4682 | . . 3 ⊢ (∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅) |
5 | 4 | abbii 2798 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅} |
6 | df-qs 8724 | . 2 ⊢ ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} | |
7 | df-sn 4625 | . 2 ⊢ {[𝐴]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅} | |
8 | 5, 6, 7 | 3eqtr4ri 2767 | 1 ⊢ {[𝐴]𝑅} = ({𝐴} / 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 {cab 2705 ∃wrex 3066 Vcvv 3470 {csn 4624 [cec 8716 / cqs 8717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ec 8720 df-qs 8724 |
This theorem is referenced by: (None) |
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