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Theorem res0 5264
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 4997 . 2  |-  ( A  |`  (/) )  =  ( A  i^i  ( (/)  X. 
_V ) )
2 0xp 5066 . . 3  |-  ( (/)  X. 
_V )  =  (/)
32ineq2i 3679 . 2  |-  ( A  i^i  ( (/)  X.  _V ) )  =  ( A  i^i  (/) )
4 in0 3793 . 2  |-  ( A  i^i  (/) )  =  (/)
51, 3, 43eqtri 2474 1  |-  ( A  |`  (/) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1381   _Vcvv 3093    i^i cin 3457   (/)c0 3767    X. cxp 4983    |` cres 4987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-opab 4492  df-xp 4991  df-res 4997
This theorem is referenced by:  ima0  5338  resdisj  5422  smo0  7027  tfrlem16  7060  tz7.44-1  7070  mapunen  7684  fnfi  7796  ackbij2lem3  8619  hashf1lem1  12478  setsid  14545  meet0  15636  join0  15637  frmdplusg  15891  psgn0fv0  16405  gsum2dlem2  16867  gsum2dOLD  16869  ablfac1eulem  16991  ablfac1eu  16992  psrplusg  17902  ply1plusgfvi  18151  ptuncnv  20174  ptcmpfi  20180  ust0  20588  xrge0gsumle  21204  xrge0tsms  21205  jensen  23183  0pth  24437  1pthonlem1  24456  eupath2  24845  zrdivrng  25299  resf1o  27418  gsumle  27636  xrge0tsmsd  27641  esumsn  27938  dfpo2  29152  eldm3  29159  rdgprc0  29194  eldioph4b  30713  diophren  30715
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