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Theorem ress0 13202
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 3483 . . 3  |-  (/)  C_  A
2 0ex 4150 . . 3  |-  (/)  e.  _V
3 eqid 2283 . . . 4  |-  ( (/)s  A )  =  ( (/)s  A )
4 base0 13185 . . . 4  |-  (/)  =  (
Base `  (/) )
53, 4ressid2 13196 . . 3  |-  ( (
(/)  C_  A  /\  (/)  e.  _V  /\  A  e.  _V )  ->  ( (/)s  A )  =  (/) )
61, 2, 5mp3an12 1267 . 2  |-  ( A  e.  _V  ->  ( (/)s  A
)  =  (/) )
7 reldmress 13194 . . 3  |-  Rel  doms
87ovprc2 5887 . 2  |-  ( -.  A  e.  _V  ->  (
(/)s  A )  =  (/) )
96, 8pm2.61i 156 1  |-  ( (/)s  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455  (class class class)co 5858   ↾s cress 13149
This theorem is referenced by:  ressress  13205  invrfval  15455  mplval  16173  ply1val  16273  dsmmval  27200  dsmmval2  27202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-slot 13152  df-base 13153  df-ress 13155
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