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Mirrors > Home > MPE Home > Th. List > elqsg | Structured version Visualization version GIF version |
Description: Closed form of elqs 8807. (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 2738 | . . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = [𝑥]𝑅 ↔ 𝐵 = [𝑥]𝑅)) | |
2 | 1 | rexbidv 3176 | . 2 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) |
3 | df-qs 8749 | . 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
4 | 2, 3 | elab2g 3682 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 [cec 8741 / cqs 8742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rex 3068 df-qs 8749 |
This theorem is referenced by: elqs 8807 elqsi 8808 elqsecl 8809 ecelqsg 8810 quselbas 19214 ghmqusnsglem2 19311 ghmquskerlem2 19315 rngqiprngfulem1 21338 elpi1 25091 eldmqsres 38268 eldmqs1cossres 38640 prtlem11 38847 |
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