| 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 6045 | . 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 2108 Vcvv 3480 ↾ cres 5687 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-res 5697 |
| This theorem is referenced by: gsum2dlem2 19989 tsmspropd 24140 ulmss 26440 elrgspnlem4 33249 lmimdim 33654 aks6d1c6lem3 42173 psrbagres 42556 pwssplit4 43101 limsupresre 45711 limsupresico 45715 limsupresuz 45718 limsupres 45720 limsupresxr 45781 liminfresxr 45782 liminfresico 45786 liminfresre 45794 liminfresuz 45799 isubgriedg 47849 isubgrvtx 47853 |
| Copyright terms: Public domain | W3C validator |