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Theorem ress0 14792
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 3765 . . 3  |-  (/)  C_  A
2 0ex 4523 . . 3  |-  (/)  e.  _V
3 eqid 2400 . . . 4  |-  ( (/)s  A )  =  ( (/)s  A )
4 base0 14772 . . . 4  |-  (/)  =  (
Base `  (/) )
53, 4ressid2 14786 . . 3  |-  ( (
(/)  C_  A  /\  (/)  e.  _V  /\  A  e.  _V )  ->  ( (/)s  A )  =  (/) )
61, 2, 5mp3an12 1314 . 2  |-  ( A  e.  _V  ->  ( (/)s  A
)  =  (/) )
7 reldmress 14784 . . 3  |-  Rel  doms
87ovprc2 6264 . 2  |-  ( -.  A  e.  _V  ->  (
(/)s  A )  =  (/) )
96, 8pm2.61i 164 1  |-  ( (/)s  A )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1403    e. wcel 1840   _Vcvv 3056    C_ wss 3411   (/)c0 3735  (class class class)co 6232   ↾s cress 14732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5487  df-fun 5525  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-slot 14735  df-base 14736  df-ress 14738
This theorem is referenced by:  ressress  14796  invrfval  17532  mplval  18294  ply1val  18443  dsmmval  18953  dsmmval2  18955  resvsca  28154
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