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Theorem snres0 6305
Description: Condition for restriction of a singleton to be empty. (Contributed by Scott Fenton, 9-Aug-2024.)
Hypothesis
Ref Expression
snres0.1 𝐵 ∈ V
Assertion
Ref Expression
snres0 (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ ¬ 𝐴𝐶)

Proof of Theorem snres0
StepHypRef Expression
1 relres 6013 . . 3 Rel ({⟨𝐴, 𝐵⟩} ↾ 𝐶)
2 reldm0 5932 . . 3 (Rel ({⟨𝐴, 𝐵⟩} ↾ 𝐶) → (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅))
31, 2ax-mp 5 . 2 (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅)
4 dmres 6019 . . . 4 dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ dom {⟨𝐴, 𝐵⟩})
5 snres0.1 . . . . . 6 𝐵 ∈ V
65dmsnop 6223 . . . . 5 dom {⟨𝐴, 𝐵⟩} = {𝐴}
76ineq2i 4209 . . . 4 (𝐶 ∩ dom {⟨𝐴, 𝐵⟩}) = (𝐶 ∩ {𝐴})
84, 7eqtri 2755 . . 3 dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ {𝐴})
98eqeq1i 2732 . 2 (dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ (𝐶 ∩ {𝐴}) = ∅)
10 disjsn 4718 . 2 ((𝐶 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝐶)
113, 9, 103bitri 296 1 (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ ¬ 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1533  wcel 2098  Vcvv 3471  cin 3946  c0 4324  {csn 4630  cop 4636  dom cdm 5680  cres 5682  Rel wrel 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-xp 5686  df-rel 5687  df-dm 5690  df-res 5692
This theorem is referenced by:  noinfbnd2lem1  27681
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