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Mirrors > Home > MPE Home > Th. List > snres0 | Structured version Visualization version GIF version |
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 6013 | . . 3 ⊢ Rel ({⟨𝐴, 𝐵⟩} ↾ 𝐶) | |
2 | reldm0 5932 | . . 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 6019 | . . . 4 ⊢ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ dom {⟨𝐴, 𝐵⟩}) | |
5 | snres0.1 | . . . . . 6 ⊢ 𝐵 ∈ V | |
6 | 5 | dmsnop 6223 | . . . . 5 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴} |
7 | 6 | ineq2i 4209 | . . . 4 ⊢ (𝐶 ∩ dom {⟨𝐴, 𝐵⟩}) = (𝐶 ∩ {𝐴}) |
8 | 4, 7 | eqtri 2755 | . . 3 ⊢ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ {𝐴}) |
9 | 8 | eqeq1i 2732 | . 2 ⊢ (dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ (𝐶 ∩ {𝐴}) = ∅) |
10 | disjsn 4718 | . 2 ⊢ ((𝐶 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐶) | |
11 | 3, 9, 10 | 3bitri 296 | 1 ⊢ (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ ¬ 𝐴 ∈ 𝐶) |
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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