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Theorem res0 4644
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 4383 . 2 (𝐴 ↾ ∅) = (𝐴 ∩ (∅ × V))
2 0xp 4446 . . 3 (∅ × V) = ∅
32ineq2i 3171 . 2 (𝐴 ∩ (∅ × V)) = (𝐴 ∩ ∅)
4 in0 3286 . 2 (𝐴 ∩ ∅) = ∅
51, 3, 43eqtri 2106 1 (𝐴 ↾ ∅) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1285  Vcvv 2602  cin 2973  c0 3258   × cxp 4369  cres 4373
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 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-opab 3848  df-xp 4377  df-res 4383
This theorem is referenced by:  ima0  4714  resdisj  4781  smo0  5947  tfr0dm  5971  tfr0  5972  fnfi  6446
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