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Mirrors > Home > ILE Home > Th. List > resex | GIF version |
Description: The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.) |
Ref | Expression |
---|---|
resex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
resex | ⊢ (𝐴 ↾ 𝐵) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | resexg 4707 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ↾ 𝐵) ∈ V) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐴 ↾ 𝐵) ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1434 Vcvv 2612 ↾ cres 4401 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2614 df-in 2990 df-ss 2997 df-res 4411 |
This theorem is referenced by: finomni 6699 |
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