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Theorem res0 4872
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 4600 . 2  |-  ( A  |`  (/) )  =  ( A  i^i  ( (/)  X. 
_V ) )
2 0xp 4668 . . 3  |-  ( (/)  X. 
_V )  =  (/)
32ineq2i 3306 . 2  |-  ( A  i^i  ( (/)  X.  _V ) )  =  ( A  i^i  (/) )
4 in0 3429 . 2  |-  ( A  i^i  (/) )  =  (/)
51, 3, 43eqtri 2182 1  |-  ( A  |`  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1335   _Vcvv 2712    i^i cin 3101   (/)c0 3395    X. cxp 4586    |` cres 4590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-opab 4028  df-xp 4594  df-res 4600
This theorem is referenced by:  ima0  4947  resdisj  5016  smo0  6247  tfr0dm  6271  tfr0  6272  fnfi  6883  setsslid  12310
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