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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 5857 | . . 3 ⊢ Rel ({〈𝐴, 𝐵〉} ↾ 𝐶) | |
2 | reldm0 5774 | . . 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 5850 | . . . 4 ⊢ dom ({〈𝐴, 𝐵〉} ↾ 𝐶) = (𝐶 ∩ dom {〈𝐴, 𝐵〉}) | |
5 | snres0.1 | . . . . . 6 ⊢ 𝐵 ∈ V | |
6 | 5 | dmsnop 6050 | . . . . 5 ⊢ dom {〈𝐴, 𝐵〉} = {𝐴} |
7 | 6 | ineq2i 4116 | . . . 4 ⊢ (𝐶 ∩ dom {〈𝐴, 𝐵〉}) = (𝐶 ∩ {𝐴}) |
8 | 4, 7 | eqtri 2781 | . . 3 ⊢ dom ({〈𝐴, 𝐵〉} ↾ 𝐶) = (𝐶 ∩ {𝐴}) |
9 | 8 | eqeq1i 2763 | . 2 ⊢ (dom ({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅ ↔ (𝐶 ∩ {𝐴}) = ∅) |
10 | disjsn 4607 | . 2 ⊢ ((𝐶 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐶) | |
11 | 3, 9, 10 | 3bitri 300 | 1 ⊢ (({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅ ↔ ¬ 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1538 ∈ wcel 2111 Vcvv 3409 ∩ cin 3859 ∅c0 4227 {csn 4525 〈cop 4531 dom cdm 5528 ↾ cres 5530 Rel wrel 5533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-br 5037 df-opab 5099 df-xp 5534 df-rel 5535 df-dm 5538 df-res 5540 |
This theorem is referenced by: noinfbnd2lem1 33530 |
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