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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 4785 | . 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 1445 Vcvv 2633 ↾ cres 4469 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-in 3019 df-ss 3026 df-res 4479 |
This theorem is referenced by: sbthlemi10 6755 finomni 6883 |
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