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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 6026 | . . 3 ⊢ Rel ({〈𝐴, 𝐵〉} ↾ 𝐶) | |
2 | reldm0 5941 | . . 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 6032 | . . . 4 ⊢ dom ({〈𝐴, 𝐵〉} ↾ 𝐶) = (𝐶 ∩ dom {〈𝐴, 𝐵〉}) | |
5 | snres0.1 | . . . . . 6 ⊢ 𝐵 ∈ V | |
6 | 5 | dmsnop 6238 | . . . . 5 ⊢ dom {〈𝐴, 𝐵〉} = {𝐴} |
7 | 6 | ineq2i 4225 | . . . 4 ⊢ (𝐶 ∩ dom {〈𝐴, 𝐵〉}) = (𝐶 ∩ {𝐴}) |
8 | 4, 7 | eqtri 2763 | . . 3 ⊢ dom ({〈𝐴, 𝐵〉} ↾ 𝐶) = (𝐶 ∩ {𝐴}) |
9 | 8 | eqeq1i 2740 | . 2 ⊢ (dom ({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅ ↔ (𝐶 ∩ {𝐴}) = ∅) |
10 | disjsn 4716 | . 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 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ∅c0 4339 {csn 4631 〈cop 4637 dom cdm 5689 ↾ cres 5691 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-dm 5699 df-res 5701 |
This theorem is referenced by: noinfbnd2lem1 27790 |
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