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Theorem ress0 13218
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 3496 . . 3  |-  (/)  C_  A
2 0ex 4166 . . 3  |-  (/)  e.  _V
3 eqid 2296 . . . 4  |-  ( (/)s  A )  =  ( (/)s  A )
4 base0 13201 . . . 4  |-  (/)  =  (
Base `  (/) )
53, 4ressid2 13212 . . 3  |-  ( (
(/)  C_  A  /\  (/)  e.  _V  /\  A  e.  _V )  ->  ( (/)s  A )  =  (/) )
61, 2, 5mp3an12 1267 . 2  |-  ( A  e.  _V  ->  ( (/)s  A
)  =  (/) )
7 reldmress 13210 . . 3  |-  Rel  doms
87ovprc2 5903 . 2  |-  ( -.  A  e.  _V  ->  (
(/)s  A )  =  (/) )
96, 8pm2.61i 156 1  |-  ( (/)s  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   (/)c0 3468  (class class class)co 5874   ↾s cress 13165
This theorem is referenced by:  ressress  13221  invrfval  15471  mplval  16189  ply1val  16289  dsmmval  27303  dsmmval2  27305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-slot 13168  df-base 13169  df-ress 13171
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