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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 8697 | . 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
2 | abrexexg 7934 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} ∈ V) | |
3 | 1, 2 | eqeltrid 2838 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {cab 2710 ∃wrex 3071 Vcvv 3475 [cec 8689 / cqs 8690 |
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 2704 ax-rep 5281 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-rex 3072 df-v 3477 df-qs 8697 |
This theorem is referenced by: qsex 8758 pstmval 32806 pstmxmet 32808 imaexALTV 37105 |
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