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Theorem ressnop0 6300
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 5061 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × V) → 𝐴𝐶)
21con3i 148 . 2 𝐴𝐶 → ¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V))
3 df-res 5037 . . . 4 ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ({⟨𝐴, 𝐵⟩} ∩ (𝐶 × V))
4 incom 3763 . . . 4 ({⟨𝐴, 𝐵⟩} ∩ (𝐶 × V)) = ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩})
53, 4eqtri 2628 . . 3 ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩})
6 disjsn 4188 . . . 4 (((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩}) = ∅ ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V))
76biimpri 216 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V) → ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩}) = ∅)
85, 7syl5eq 2652 . 2 (¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V) → ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅)
92, 8syl 17 1 𝐴𝐶 → ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1474  wcel 1976  Vcvv 3169  cin 3535  c0 3870  {csn 4121  cop 4127   × cxp 5023  cres 5027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-opab 4635  df-xp 5031  df-res 5037
This theorem is referenced by:  fvunsn  6325  fsnunres  6334  wfrlem14  7289  constr3pthlem1  25946  ex-res  26453
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