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Mirrors > Home > MPE Home > Th. List > Mathboxes > resexd | Structured version Visualization version GIF version |
Description: The restriction of a set is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
resexd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
resexd | ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resexd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | resexg 5892 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Vcvv 3494 ↾ cres 5551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3942 df-ss 3951 df-res 5561 |
This theorem is referenced by: limsupresre 41970 limsupresico 41974 limsupresuz 41977 limsupres 41979 limsupresxr 42040 liminfresxr 42041 liminfresico 42045 liminfresre 42053 liminfresuz 42058 |
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