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Theorem ressnop0 5661
Description: If 𝐴 is not in 𝐶, then the restriction of a singleton of 𝐴, 𝐵 to 𝐶 is null. (Contributed by Scott Fenton, 15-Apr-2011.)
Assertion
Ref Expression
ressnop0 𝐴𝐶 → ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅)

Proof of Theorem ressnop0
StepHypRef Expression
1 opelxp1 4633 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × V) → 𝐴𝐶)
21con3i 622 . 2 𝐴𝐶 → ¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V))
3 df-res 4611 . . . 4 ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ({⟨𝐴, 𝐵⟩} ∩ (𝐶 × V))
4 incom 3310 . . . 4 ({⟨𝐴, 𝐵⟩} ∩ (𝐶 × V)) = ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩})
53, 4eqtri 2185 . . 3 ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩})
6 disjsn 3633 . . . 4 (((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩}) = ∅ ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V))
76biimpri 132 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V) → ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩}) = ∅)
85, 7syl5eq 2209 . 2 (¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V) → ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅)
92, 8syl 14 1 𝐴𝐶 → ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1342  wcel 2135  Vcvv 2722  cin 3111  c0 3405  {csn 3571  cop 3574   × cxp 4597  cres 4601
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-opab 4039  df-xp 4605  df-res 4611
This theorem is referenced by:  fvunsng  5674  fsnunres  5682
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