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Mirrors > Home > MPE Home > Th. List > qsexg | Structured version Visualization version GIF version |
Description: A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
qsexg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-qs 8730 | . 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
2 | abrexexg 7964 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} ∈ V) | |
3 | 1, 2 | eqeltrid 2833 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {cab 2705 ∃wrex 3067 Vcvv 3471 [cec 8722 / cqs 8723 |
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 ax-rep 5285 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-mo 2530 df-clab 2706 df-cleq 2720 df-clel 2806 df-rex 3068 df-v 3473 df-qs 8730 |
This theorem is referenced by: qsex 8794 pstmval 33496 pstmxmet 33498 imaexALTV 37802 |
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