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Theorem ress0 13523
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 3656 . . 3  |-  (/)  C_  A
2 0ex 4339 . . 3  |-  (/)  e.  _V
3 eqid 2436 . . . 4  |-  ( (/)s  A )  =  ( (/)s  A )
4 base0 13506 . . . 4  |-  (/)  =  (
Base `  (/) )
53, 4ressid2 13517 . . 3  |-  ( (
(/)  C_  A  /\  (/)  e.  _V  /\  A  e.  _V )  ->  ( (/)s  A )  =  (/) )
61, 2, 5mp3an12 1269 . 2  |-  ( A  e.  _V  ->  ( (/)s  A
)  =  (/) )
7 reldmress 13515 . . 3  |-  Rel  doms
87ovprc2 6110 . 2  |-  ( -.  A  e.  _V  ->  (
(/)s  A )  =  (/) )
96, 8pm2.61i 158 1  |-  ( (/)s  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2956    C_ wss 3320   (/)c0 3628  (class class class)co 6081   ↾s cress 13470
This theorem is referenced by:  ressress  13526  invrfval  15778  mplval  16492  ply1val  16592  dsmmval  27177  dsmmval2  27179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-slot 13473  df-base 13474  df-ress 13476
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