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Theorem res0 4932
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 4659 . 2 (𝐴 ↾ ∅) = (𝐴 ∩ (∅ × V))
2 0xp 4727 . . 3 (∅ × V) = ∅
32ineq2i 3348 . 2 (𝐴 ∩ (∅ × V)) = (𝐴 ∩ ∅)
4 in0 3472 . 2 (𝐴 ∩ ∅) = ∅
51, 3, 43eqtri 2214 1 (𝐴 ↾ ∅) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1364  Vcvv 2752  cin 3143  c0 3437   × cxp 4645  cres 4649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-opab 4083  df-xp 4653  df-res 4659
This theorem is referenced by:  ima0  5008  resdisj  5078  smo0  6327  tfr0dm  6351  tfr0  6352  fnfi  6970  setsslid  12574
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