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Theorem ressnop0 6566
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 5289 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × V) → 𝐴𝐶)
21con3i 151 . 2 𝐴𝐶 → ¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V))
3 df-res 5262 . . . 4 ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ({⟨𝐴, 𝐵⟩} ∩ (𝐶 × V))
4 incom 3956 . . . 4 ({⟨𝐴, 𝐵⟩} ∩ (𝐶 × V)) = ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩})
53, 4eqtri 2793 . . 3 ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩})
6 disjsn 4384 . . . 4 (((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩}) = ∅ ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V))
76biimpri 218 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V) → ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩}) = ∅)
85, 7syl5eq 2817 . 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 1631  wcel 2145  Vcvv 3351  cin 3722  c0 4063  {csn 4317  cop 4323   × cxp 5248  cres 5252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-opab 4848  df-xp 5256  df-res 5262
This theorem is referenced by:  fvunsn  6592  fsnunres  6601  wfrlem14  7585  ex-res  27640
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