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Theorem res0 4975
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 4717 . 2  |-  ( A  |`  (/) )  =  ( A  i^i  ( (/)  X. 
_V ) )
2 xp0r 4784 . . 3  |-  ( (/)  X. 
_V )  =  (/)
32ineq2i 3380 . 2  |-  ( A  i^i  ( (/)  X.  _V ) )  =  ( A  i^i  (/) )
4 in0 3493 . 2  |-  ( A  i^i  (/) )  =  (/)
51, 3, 43eqtri 2320 1  |-  ( A  |`  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1632   _Vcvv 2801    i^i cin 3164   (/)c0 3468    X. cxp 4703    |` cres 4707
This theorem is referenced by:  ima0  5046  resdisj  5121  smo0  6391  tfrlem16  6425  tz7.44-1  6435  mapunen  7046  fnfi  7150  ackbij2lem3  7883  hashf1lem1  11409  setsid  13203  frmdplusg  14492  gsum2d  15239  ablfac1eulem  15323  ablfac1eu  15324  psrplusg  16142  ply1plusgfvi  16336  ptuncnv  17514  ptcmpfi  17520  xrge0gsumle  18354  xrge0tsms  18355  jensen  20299  zrdivrng  21115  xrge0tsmsd  23397  esumsn  23452  eupath2  23919  dfpo2  24183  eldm3  24190  rdgprc0  24221  empos  25345  eldioph4b  26997  diophren  26999  0pth  28356
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  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-opab 4094  df-xp 4711  df-res 4717
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