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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 4397 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
3 | eqid 2728 | . . . 4 ⊢ (∅ ↾s 𝐴) = (∅ ↾s 𝐴) | |
4 | base0 17184 | . . . 4 ⊢ ∅ = (Base‘∅) | |
5 | 3, 4 | ressid2 17212 | . . 3 ⊢ ((∅ ⊆ 𝐴 ∧ ∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾s 𝐴) = ∅) |
6 | 1, 2, 5 | mp3an12 1448 | . 2 ⊢ (𝐴 ∈ V → (∅ ↾s 𝐴) = ∅) |
7 | reldmress 17210 | . . 3 ⊢ Rel dom ↾s | |
8 | 7 | ovprc2 7460 | . 2 ⊢ (¬ 𝐴 ∈ V → (∅ ↾s 𝐴) = ∅) |
9 | 6, 8 | pm2.61i 182 | 1 ⊢ (∅ ↾s 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⊆ wss 3947 ∅c0 4323 (class class class)co 7420 ↾s cress 17208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-1cn 11196 ax-addcl 11198 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-nn 12243 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 |
This theorem is referenced by: ressress 17228 symgval 19322 invrfval 20327 dsmmval 21667 dsmmval2 21669 mplval 21930 ply1val 22112 resvsca 33041 |
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