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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 8655 | . 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
2 | abrexexg 7894 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} ∈ V) | |
3 | 1, 2 | eqeltrid 2842 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {cab 2714 ∃wrex 3074 Vcvv 3446 [cec 8647 / cqs 8648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-rep 5243 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-mo 2539 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-v 3448 df-qs 8655 |
This theorem is referenced by: qsex 8716 pstmval 32479 pstmxmet 32481 imaexALTV 36794 |
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