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Theorem ress0 16549
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 4322 . . 3 ∅ ⊆ 𝐴
2 0ex 5187 . . 3 ∅ ∈ V
3 eqid 2822 . . . 4 (∅ ↾s 𝐴) = (∅ ↾s 𝐴)
4 base0 16527 . . . 4 ∅ = (Base‘∅)
53, 4ressid2 16543 . . 3 ((∅ ⊆ 𝐴 ∧ ∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾s 𝐴) = ∅)
61, 2, 5mp3an12 1448 . 2 (𝐴 ∈ V → (∅ ↾s 𝐴) = ∅)
7 reldmress 16541 . . 3 Rel dom ↾s
87ovprc2 7180 . 2 𝐴 ∈ V → (∅ ↾s 𝐴) = ∅)
96, 8pm2.61i 185 1 (∅ ↾s 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2114  Vcvv 3469  wss 3908  c0 4265  (class class class)co 7140  s cress 16475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-slot 16478  df-base 16480  df-ress 16482
This theorem is referenced by:  ressress  16553  symgval  18488  invrfval  19417  dsmmval  20421  dsmmval2  20423  mplval  20664  ply1val  20821  resvsca  30935
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