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Mirrors > Home > ILE Home > Th. List > dmin | GIF version |
Description: The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.) |
Ref | Expression |
---|---|
dmin | ⊢ dom (𝐴 ∩ 𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.40 1610 | . . 3 ⊢ (∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) → (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) | |
2 | vex 2689 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | eldm2 4737 | . . . 4 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵)) |
4 | elin 3259 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
5 | 4 | exbii 1584 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
6 | 3, 5 | bitri 183 | . . 3 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
7 | elin 3259 | . . . 4 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵)) | |
8 | 2 | eldm2 4737 | . . . . 5 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
9 | 2 | eldm2 4737 | . . . . 5 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵) |
10 | 8, 9 | anbi12i 455 | . . . 4 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) |
11 | 7, 10 | bitri 183 | . . 3 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) |
12 | 1, 6, 11 | 3imtr4i 200 | . 2 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) → 𝑥 ∈ (dom 𝐴 ∩ dom 𝐵)) |
13 | 12 | ssriv 3101 | 1 ⊢ dom (𝐴 ∩ 𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∃wex 1468 ∈ wcel 1480 ∩ cin 3070 ⊆ wss 3071 〈cop 3530 dom cdm 4539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-dm 4549 |
This theorem is referenced by: rnin 4948 |
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