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Theorem ressnop0 5287
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 A 𝐶 → ({⟨A, B⟩} ↾ 𝐶) = ∅)

Proof of Theorem ressnop0
StepHypRef Expression
1 opelxp1 4320 . . 3 (⟨A, B (𝐶 × V) → A 𝐶)
21con3i 561 . 2 A 𝐶 → ¬ ⟨A, B (𝐶 × V))
3 df-res 4300 . . . 4 ({⟨A, B⟩} ↾ 𝐶) = ({⟨A, B⟩} ∩ (𝐶 × V))
4 incom 3123 . . . 4 ({⟨A, B⟩} ∩ (𝐶 × V)) = ((𝐶 × V) ∩ {⟨A, B⟩})
53, 4eqtri 2057 . . 3 ({⟨A, B⟩} ↾ 𝐶) = ((𝐶 × V) ∩ {⟨A, B⟩})
6 disjsn 3423 . . . 4 (((𝐶 × V) ∩ {⟨A, B⟩}) = ∅ ↔ ¬ ⟨A, B (𝐶 × V))
76biimpri 124 . . 3 (¬ ⟨A, B (𝐶 × V) → ((𝐶 × V) ∩ {⟨A, B⟩}) = ∅)
85, 7syl5eq 2081 . 2 (¬ ⟨A, B (𝐶 × V) → ({⟨A, B⟩} ↾ 𝐶) = ∅)
92, 8syl 14 1 A 𝐶 → ({⟨A, B⟩} ↾ 𝐶) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1242   wcel 1390  Vcvv 2551  cin 2910  c0 3218  {csn 3367  cop 3370   × cxp 4286  cres 4290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294  df-res 4300
This theorem is referenced by:  fvunsng  5300  fsnunres  5307
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