![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 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 5864 | . 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 2111 Vcvv 3441 ↾ cres 5521 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 ax-sep 5167 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-in 3888 df-ss 3898 df-res 5531 |
This theorem is referenced by: limsupresre 42338 limsupresico 42342 limsupresuz 42345 limsupres 42347 limsupresxr 42408 liminfresxr 42409 liminfresico 42413 liminfresre 42421 liminfresuz 42426 |
Copyright terms: Public domain | W3C validator |