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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 6047 | . 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 2106 Vcvv 3478 ↾ cres 5691 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-in 3970 df-ss 3980 df-res 5701 |
This theorem is referenced by: gsum2dlem2 20004 tsmspropd 24156 ulmss 26455 elrgspnlem4 33235 lmimdim 33631 aks6d1c6lem3 42154 psrbagres 42533 pwssplit4 43078 limsupresre 45652 limsupresico 45656 limsupresuz 45659 limsupres 45661 limsupresxr 45722 liminfresxr 45723 liminfresico 45727 liminfresre 45735 liminfresuz 45740 isubgriedg 47787 isubgrvtx 47791 |
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