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Theorem res0 5142
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 4882 . 2  |-  ( A  |`  (/) )  =  ( A  i^i  ( (/)  X. 
_V ) )
2 xp0r 4948 . . 3  |-  ( (/)  X. 
_V )  =  (/)
32ineq2i 3531 . 2  |-  ( A  i^i  ( (/)  X.  _V ) )  =  ( A  i^i  (/) )
4 in0 3645 . 2  |-  ( A  i^i  (/) )  =  (/)
51, 3, 43eqtri 2459 1  |-  ( A  |`  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1652   _Vcvv 2948    i^i cin 3311   (/)c0 3620    X. cxp 4868    |` cres 4872
This theorem is referenced by:  ima0  5213  resdisj  5290  smo0  6612  tfrlem16  6646  tz7.44-1  6656  mapunen  7268  fnfi  7376  ackbij2lem3  8113  hashf1lem1  11696  setsid  13500  frmdplusg  14791  gsum2d  15538  ablfac1eulem  15622  ablfac1eu  15623  psrplusg  16437  ply1plusgfvi  16628  ptuncnv  17831  ptcmpfi  17837  ust0  18241  xrge0gsumle  18856  xrge0tsms  18857  jensen  20819  0pth  21562  1pthonlem1  21581  eupath2  21694  zrdivrng  22012  xrge0tsmsd  24215  esumsn  24448  dfpo2  25370  eldm3  25377  rdgprc0  25413  eldioph4b  26863  diophren  26865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259  df-xp 4876  df-res 4882
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