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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 8705 | . 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
2 | abrexexg 7943 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} ∈ V) | |
3 | 1, 2 | eqeltrid 2837 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {cab 2709 ∃wrex 3070 Vcvv 3474 [cec 8697 / cqs 8698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-rep 5284 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-mo 2534 df-clab 2710 df-cleq 2724 df-clel 2810 df-rex 3071 df-v 3476 df-qs 8705 |
This theorem is referenced by: qsex 8766 pstmval 32863 pstmxmet 32865 imaexALTV 37187 |
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