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Mirrors > Home > MPE Home > Th. List > ress0 | Structured version Visualization version GIF version |
Description: All restrictions of the null set are trivial. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
ress0 | ⊢ (∅ ↾s 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4395 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | 0ex 5306 | . . 3 ⊢ ∅ ∈ V | |
3 | eqid 2732 | . . . 4 ⊢ (∅ ↾s 𝐴) = (∅ ↾s 𝐴) | |
4 | base0 17145 | . . . 4 ⊢ ∅ = (Base‘∅) | |
5 | 3, 4 | ressid2 17173 | . . 3 ⊢ ((∅ ⊆ 𝐴 ∧ ∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾s 𝐴) = ∅) |
6 | 1, 2, 5 | mp3an12 1451 | . 2 ⊢ (𝐴 ∈ V → (∅ ↾s 𝐴) = ∅) |
7 | reldmress 17171 | . . 3 ⊢ Rel dom ↾s | |
8 | 7 | ovprc2 7445 | . 2 ⊢ (¬ 𝐴 ∈ V → (∅ ↾s 𝐴) = ∅) |
9 | 6, 8 | pm2.61i 182 | 1 ⊢ (∅ ↾s 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3947 ∅c0 4321 (class class class)co 7405 ↾s cress 17169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-1cn 11164 ax-addcl 11166 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-nn 12209 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 |
This theorem is referenced by: ressress 17189 symgval 19230 invrfval 20195 dsmmval 21280 dsmmval2 21282 mplval 21539 ply1val 21709 resvsca 32432 |
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