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Theorem res0 4923
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 4650 . 2 (𝐴 ↾ ∅) = (𝐴 ∩ (∅ × V))
2 0xp 4718 . . 3 (∅ × V) = ∅
32ineq2i 3345 . 2 (𝐴 ∩ (∅ × V)) = (𝐴 ∩ ∅)
4 in0 3469 . 2 (𝐴 ∩ ∅) = ∅
51, 3, 43eqtri 2212 1 (𝐴 ↾ ∅) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1363  Vcvv 2749  cin 3140  c0 3434   × cxp 4636  cres 4640
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-opab 4077  df-xp 4644  df-res 4650
This theorem is referenced by:  ima0  4999  resdisj  5069  smo0  6313  tfr0dm  6337  tfr0  6338  fnfi  6950  setsslid  12527
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