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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 4394 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | 0ex 5303 | . . 3 ⊢ ∅ ∈ V | |
3 | eqid 2733 | . . . 4 ⊢ (∅ ↾s 𝐴) = (∅ ↾s 𝐴) | |
4 | base0 17136 | . . . 4 ⊢ ∅ = (Base‘∅) | |
5 | 3, 4 | ressid2 17164 | . . 3 ⊢ ((∅ ⊆ 𝐴 ∧ ∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾s 𝐴) = ∅) |
6 | 1, 2, 5 | mp3an12 1452 | . 2 ⊢ (𝐴 ∈ V → (∅ ↾s 𝐴) = ∅) |
7 | reldmress 17162 | . . 3 ⊢ Rel dom ↾s | |
8 | 7 | ovprc2 7436 | . 2 ⊢ (¬ 𝐴 ∈ V → (∅ ↾s 𝐴) = ∅) |
9 | 6, 8 | pm2.61i 182 | 1 ⊢ (∅ ↾s 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3946 ∅c0 4320 (class class class)co 7396 ↾s cress 17160 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-1cn 11155 ax-addcl 11157 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-nn 12200 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 |
This theorem is referenced by: ressress 17180 symgval 19220 invrfval 20181 dsmmval 21262 dsmmval2 21264 mplval 21519 ply1val 21687 resvsca 32406 |
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