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| Mirrors > Home > MPE Home > Th. List > elqsg | Structured version Visualization version GIF version | ||
| Description: Closed form of elqs 8809. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
| Ref | Expression |
|---|---|
| elqsg | ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqeq1 2741 | . . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = [𝑥]𝑅 ↔ 𝐵 = [𝑥]𝑅)) | |
| 2 | 1 | rexbidv 3179 | . 2 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) |
| 3 | df-qs 8751 | . 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
| 4 | 2, 3 | elab2g 3680 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 [cec 8743 / cqs 8744 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-qs 8751 |
| This theorem is referenced by: elqs 8809 elqsi 8810 elqsecl 8811 ecelqsg 8812 quselbas 19202 ghmqusnsglem2 19299 ghmquskerlem2 19303 rngqiprngfulem1 21321 elpi1 25078 eldmqsres 38288 eldmqs1cossres 38660 prtlem11 38867 |
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