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Theorem res0 4673
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 4411 . 2  |-  ( A  |`  (/) )  =  ( A  i^i  ( (/)  X. 
_V ) )
2 0xp 4474 . . 3  |-  ( (/)  X. 
_V )  =  (/)
32ineq2i 3182 . 2  |-  ( A  i^i  ( (/)  X.  _V ) )  =  ( A  i^i  (/) )
4 in0 3300 . 2  |-  ( A  i^i  (/) )  =  (/)
51, 3, 43eqtri 2107 1  |-  ( A  |`  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1285   _Vcvv 2612    i^i cin 2983   (/)c0 3269    X. cxp 4397    |` cres 4401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-opab 3866  df-xp 4405  df-res 4411
This theorem is referenced by:  ima0  4744  resdisj  4811  smo0  5993  tfr0dm  6017  tfr0  6018  fnfi  6569
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