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Mirrors > Home > MPE Home > Th. List > resexg | Structured version Visualization version 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 6007 | . 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
2 | ssexg 5324 | . 2 ⊢ (((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝐴 ↾ 𝐵) ∈ V) | |
3 | 1, 2 | mpan 689 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3949 ↾ cres 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-in 3956 df-ss 3966 df-res 5689 |
This theorem is referenced by: resexd 6029 resex 6030 fvtresfn 7001 offres 7970 ressuppss 8168 ressuppssdif 8170 resixp 8927 dif1enlem 9156 dif1enlemOLD 9157 sbthfilem 9201 fsuppres 9388 climres 15519 setsvalg 17099 setsid 17141 symgfixels 19302 qtopres 23202 vtxdginducedm1 28831 redwlk 28960 hhssva 30541 hhsssm 30542 hhshsslem1 30551 resf1o 31986 eulerpartlemmf 33405 exidres 36794 exidresid 36795 xrnresex 37324 unidmqs 37572 lmhmlnmsplit 41877 climresdm 44614 setsv 46094 |
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