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Theorem res0 4818
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 4546 . 2  |-  ( A  |`  (/) )  =  ( A  i^i  ( (/)  X. 
_V ) )
2 0xp 4614 . . 3  |-  ( (/)  X. 
_V )  =  (/)
32ineq2i 3269 . 2  |-  ( A  i^i  ( (/)  X.  _V ) )  =  ( A  i^i  (/) )
4 in0 3392 . 2  |-  ( A  i^i  (/) )  =  (/)
51, 3, 43eqtri 2162 1  |-  ( A  |`  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1331   _Vcvv 2681    i^i cin 3065   (/)c0 3358    X. cxp 4532    |` cres 4536
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-opab 3985  df-xp 4540  df-res 4546
This theorem is referenced by:  ima0  4893  resdisj  4962  smo0  6188  tfr0dm  6212  tfr0  6213  fnfi  6818  setsslid  11998
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