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Theorem ress0 15855
Description: All restrictions of the null set are trivial. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
ress0 (∅ ↾s 𝐴) = ∅

Proof of Theorem ress0
StepHypRef Expression
1 0ss 3944 . . 3 ∅ ⊆ 𝐴
2 0ex 4750 . . 3 ∅ ∈ V
3 eqid 2621 . . . 4 (∅ ↾s 𝐴) = (∅ ↾s 𝐴)
4 base0 15833 . . . 4 ∅ = (Base‘∅)
53, 4ressid2 15849 . . 3 ((∅ ⊆ 𝐴 ∧ ∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾s 𝐴) = ∅)
61, 2, 5mp3an12 1411 . 2 (𝐴 ∈ V → (∅ ↾s 𝐴) = ∅)
7 reldmress 15847 . . 3 Rel dom ↾s
87ovprc2 6638 . 2 𝐴 ∈ V → (∅ ↾s 𝐴) = ∅)
96, 8pm2.61i 176 1 (∅ ↾s 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3186  wss 3555  c0 3891  (class class class)co 6604  s cress 15782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-slot 15785  df-base 15786  df-ress 15788
This theorem is referenced by:  ressress  15859  invrfval  18594  mplval  19347  ply1val  19483  dsmmval  19997  dsmmval2  19999  resvsca  29615
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