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Mirrors > Home > ILE Home > Th. List > resexg | GIF version |
Description: The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
resexg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resss 4902 | . 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
2 | ssexg 4115 | . 2 ⊢ (((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝐴 ↾ 𝐵) ∈ V) | |
3 | 1, 2 | mpan 421 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 Vcvv 2721 ⊆ wss 3111 ↾ cres 4600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4094 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-in 3117 df-ss 3124 df-res 4610 |
This theorem is referenced by: resex 4919 offres 6095 resixp 6690 climres 11230 setsvalg 12361 setsex 12363 setsslid 12381 |
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