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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 6015 | . . 3 ⊢ Rel ({〈𝐴, 𝐵〉} ↾ 𝐶) | |
2 | reldm0 5934 | . . 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 6021 | . . . 4 ⊢ dom ({〈𝐴, 𝐵〉} ↾ 𝐶) = (𝐶 ∩ dom {〈𝐴, 𝐵〉}) | |
5 | snres0.1 | . . . . . 6 ⊢ 𝐵 ∈ V | |
6 | 5 | dmsnop 6225 | . . . . 5 ⊢ dom {〈𝐴, 𝐵〉} = {𝐴} |
7 | 6 | ineq2i 4211 | . . . 4 ⊢ (𝐶 ∩ dom {〈𝐴, 𝐵〉}) = (𝐶 ∩ {𝐴}) |
8 | 4, 7 | eqtri 2756 | . . 3 ⊢ dom ({〈𝐴, 𝐵〉} ↾ 𝐶) = (𝐶 ∩ {𝐴}) |
9 | 8 | eqeq1i 2733 | . 2 ⊢ (dom ({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅ ↔ (𝐶 ∩ {𝐴}) = ∅) |
10 | disjsn 4720 | . 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 3473 ∩ cin 3948 ∅c0 4326 {csn 4632 〈cop 4638 dom cdm 5682 ↾ cres 5684 Rel wrel 5687 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
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 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-dm 5692 df-res 5694 |
This theorem is referenced by: noinfbnd2lem1 27683 |
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