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Theorem ress0 16546
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 4347 . . 3 ∅ ⊆ 𝐴
2 0ex 5202 . . 3 ∅ ∈ V
3 eqid 2818 . . . 4 (∅ ↾s 𝐴) = (∅ ↾s 𝐴)
4 base0 16524 . . . 4 ∅ = (Base‘∅)
53, 4ressid2 16540 . . 3 ((∅ ⊆ 𝐴 ∧ ∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾s 𝐴) = ∅)
61, 2, 5mp3an12 1442 . 2 (𝐴 ∈ V → (∅ ↾s 𝐴) = ∅)
7 reldmress 16538 . . 3 Rel dom ↾s
87ovprc2 7185 . 2 𝐴 ∈ V → (∅ ↾s 𝐴) = ∅)
96, 8pm2.61i 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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