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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 4347 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | 0ex 5202 | . . 3 ⊢ ∅ ∈ V | |
3 | eqid 2818 | . . . 4 ⊢ (∅ ↾s 𝐴) = (∅ ↾s 𝐴) | |
4 | base0 16524 | . . . 4 ⊢ ∅ = (Base‘∅) | |
5 | 3, 4 | ressid2 16540 | . . 3 ⊢ ((∅ ⊆ 𝐴 ∧ ∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾s 𝐴) = ∅) |
6 | 1, 2, 5 | mp3an12 1442 | . 2 ⊢ (𝐴 ∈ V → (∅ ↾s 𝐴) = ∅) |
7 | reldmress 16538 | . . 3 ⊢ Rel dom ↾s | |
8 | 7 | ovprc2 7185 | . 2 ⊢ (¬ 𝐴 ∈ V → (∅ ↾s 𝐴) = ∅) |
9 | 6, 8 | pm2.61i 183 | 1 ⊢ (∅ ↾s 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 ∅c0 4288 (class class class)co 7145 ↾s cress 16472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-slot 16475 df-base 16477 df-ress 16479 |
This theorem is referenced by: ressress 16550 invrfval 19352 mplval 20136 ply1val 20290 dsmmval 20806 dsmmval2 20808 resvsca 30830 |
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