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Theorem ress0 14354
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 3777 . . 3  |-  (/)  C_  A
2 0ex 4533 . . 3  |-  (/)  e.  _V
3 eqid 2454 . . . 4  |-  ( (/)s  A )  =  ( (/)s  A )
4 base0 14334 . . . 4  |-  (/)  =  (
Base `  (/) )
53, 4ressid2 14348 . . 3  |-  ( (
(/)  C_  A  /\  (/)  e.  _V  /\  A  e.  _V )  ->  ( (/)s  A )  =  (/) )
61, 2, 5mp3an12 1305 . 2  |-  ( A  e.  _V  ->  ( (/)s  A
)  =  (/) )
7 reldmress 14346 . . 3  |-  Rel  doms
87ovprc2 6232 . 2  |-  ( -.  A  e.  _V  ->  (
(/)s  A )  =  (/) )
96, 8pm2.61i 164 1  |-  ( (/)s  A )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3439   (/)c0 3748  (class class class)co 6203   ↾s cress 14296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-slot 14299  df-base 14300  df-ress 14302
This theorem is referenced by:  ressress  14357  invrfval  16891  mplval  17628  ply1val  17777  dsmmval  18287  dsmmval2  18289  resvsca  26463
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