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| 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 4400 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
| 2 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
| 3 | eqid 2737 | . . . 4 ⊢ (∅ ↾s 𝐴) = (∅ ↾s 𝐴) | |
| 4 | base0 17252 | . . . 4 ⊢ ∅ = (Base‘∅) | |
| 5 | 3, 4 | ressid2 17278 | . . 3 ⊢ ((∅ ⊆ 𝐴 ∧ ∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾s 𝐴) = ∅) | 
| 6 | 1, 2, 5 | mp3an12 1453 | . 2 ⊢ (𝐴 ∈ V → (∅ ↾s 𝐴) = ∅) | 
| 7 | reldmress 17276 | . . 3 ⊢ Rel dom ↾s | |
| 8 | 7 | ovprc2 7471 | . 2 ⊢ (¬ 𝐴 ∈ V → (∅ ↾s 𝐴) = ∅) | 
| 9 | 6, 8 | pm2.61i 182 | 1 ⊢ (∅ ↾s 𝐴) = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 (class class class)co 7431 ↾s cress 17274 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 | 
| This theorem is referenced by: ressress 17293 symgval 19388 invrfval 20389 dsmmval 21754 dsmmval2 21756 mplval 22009 ply1val 22195 resvsca 33356 | 
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