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Theorem res0 5278
Description: A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
Assertion
Ref Expression
res0  |-  ( A  |`  (/) )  =  (/)

Proof of Theorem res0
StepHypRef Expression
1 df-res 5011 . 2  |-  ( A  |`  (/) )  =  ( A  i^i  ( (/)  X. 
_V ) )
2 0xp 5080 . . 3  |-  ( (/)  X. 
_V )  =  (/)
32ineq2i 3697 . 2  |-  ( A  i^i  ( (/)  X.  _V ) )  =  ( A  i^i  (/) )
4 in0 3811 . 2  |-  ( A  i^i  (/) )  =  (/)
51, 3, 43eqtri 2500 1  |-  ( A  |`  (/) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   _Vcvv 3113    i^i cin 3475   (/)c0 3785    X. cxp 4997    |` cres 5001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-xp 5005  df-res 5011
This theorem is referenced by:  ima0  5352  resdisj  5436  smo0  7030  tfrlem16  7063  tz7.44-1  7073  mapunen  7687  fnfi  7799  ackbij2lem3  8622  hashf1lem1  12471  setsid  14534  meet0  15627  join0  15628  frmdplusg  15857  psgn0fv0  16351  gsum2dlem2  16813  gsum2dOLD  16815  ablfac1eulem  16937  ablfac1eu  16938  psrplusg  17845  ply1plusgfvi  18094  ptuncnv  20135  ptcmpfi  20141  ust0  20549  xrge0gsumle  21165  xrge0tsms  21166  jensen  23143  0pth  24345  1pthonlem1  24364  eupath2  24753  zrdivrng  25207  resf1o  27322  gsumle  27530  xrge0tsmsd  27535  esumsn  27823  dfpo2  29037  eldm3  29044  rdgprc0  29079  eldioph4b  30576  diophren  30578
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