Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 5898 | . 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 2114 Vcvv 3494 ↾ cres 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3943 df-ss 3952 df-res 5567 |
This theorem is referenced by: limsupresre 41997 limsupresico 42001 limsupresuz 42004 limsupres 42006 limsupresxr 42067 liminfresxr 42068 liminfresico 42072 liminfresre 42080 liminfresuz 42085 |
Copyright terms: Public domain | W3C validator |