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| Description: Condition for restriction of a singleton to be empty. (Contributed by Scott Fenton, 9-Aug-2024.) | 
| Ref | Expression | 
|---|---|
| snres0.1 | ⊢ 𝐵 ∈ V | 
| Ref | Expression | 
|---|---|
| snres0 | ⊢ (({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅ ↔ ¬ 𝐴 ∈ 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | relres 6022 | . . 3 ⊢ Rel ({〈𝐴, 𝐵〉} ↾ 𝐶) | |
| 2 | reldm0 5937 | . . 3 ⊢ (Rel ({〈𝐴, 𝐵〉} ↾ 𝐶) → (({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅ ↔ dom ({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅)) | |
| 3 | 1, 2 | ax-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 | |
| 6 | 5 | dmsnop 6235 | . . . . 5 ⊢ dom {〈𝐴, 𝐵〉} = {𝐴} | 
| 7 | 6 | ineq2i 4216 | . . . 4 ⊢ (𝐶 ∩ dom {〈𝐴, 𝐵〉}) = (𝐶 ∩ {𝐴}) | 
| 8 | 4, 7 | eqtri 2764 | . . 3 ⊢ dom ({〈𝐴, 𝐵〉} ↾ 𝐶) = (𝐶 ∩ {𝐴}) | 
| 9 | 8 | eqeq1i 2741 | . 2 ⊢ (dom ({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅ ↔ (𝐶 ∩ {𝐴}) = ∅) | 
| 10 | disjsn 4710 | . 2 ⊢ ((𝐶 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐶) | |
| 11 | 3, 9, 10 | 3bitri 297 | 1 ⊢ (({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅ ↔ ¬ 𝐴 ∈ 𝐶) | 
| 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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