MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ress0 Structured version   Visualization version   GIF version

Theorem ress0 16141
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 4117 . . 3 ∅ ⊆ 𝐴
2 0ex 4925 . . 3 ∅ ∈ V
3 eqid 2771 . . . 4 (∅ ↾s 𝐴) = (∅ ↾s 𝐴)
4 base0 16119 . . . 4 ∅ = (Base‘∅)
53, 4ressid2 16135 . . 3 ((∅ ⊆ 𝐴 ∧ ∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾s 𝐴) = ∅)
61, 2, 5mp3an12 1562 . 2 (𝐴 ∈ V → (∅ ↾s 𝐴) = ∅)
7 reldmress 16133 . . 3 Rel dom ↾s
87ovprc2 6834 . 2 𝐴 ∈ V → (∅ ↾s 𝐴) = ∅)
96, 8pm2.61i 176 1 (∅ ↾s 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  Vcvv 3351  wss 3723  c0 4063  (class class class)co 6796  s cress 16065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-slot 16068  df-base 16070  df-ress 16072
This theorem is referenced by:  ressress  16146  invrfval  18881  mplval  19643  ply1val  19779  dsmmval  20295  dsmmval2  20297  resvsca  30170
  Copyright terms: Public domain W3C validator