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Theorem snres0 6317
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 6022 . . 3 Rel ({⟨𝐴, 𝐵⟩} ↾ 𝐶)
2 reldm0 5937 . . 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 6029 . . . 4 dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ dom {⟨𝐴, 𝐵⟩})
5 snres0.1 . . . . . 6 𝐵 ∈ V
65dmsnop 6235 . . . . 5 dom {⟨𝐴, 𝐵⟩} = {𝐴}
76ineq2i 4216 . . . 4 (𝐶 ∩ dom {⟨𝐴, 𝐵⟩}) = (𝐶 ∩ {𝐴})
84, 7eqtri 2764 . . 3 dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ {𝐴})
98eqeq1i 2741 . 2 (dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ (𝐶 ∩ {𝐴}) = ∅)
10 disjsn 4710 . 2 ((𝐶 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝐶)
113, 9, 103bitri 297 1 (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ ¬ 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1539  wcel 2107  Vcvv 3479  cin 3949  c0 4332  {csn 4625  cop 4631  dom cdm 5684  cres 5686  Rel wrel 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-dm 5694  df-res 5696
This theorem is referenced by:  noinfbnd2lem1  27776
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