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Mirrors > Home > MPE Home > Th. List > dmin | Structured version Visualization version 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 1785 | . . 3 ⊢ (∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) → (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) | |
2 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | eldm2 5244 | . . . 4 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵)) |
4 | elin 3758 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
5 | 4 | exbii 1764 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
6 | 3, 5 | bitri 263 | . . 3 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
7 | elin 3758 | . . . 4 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵)) | |
8 | 2 | eldm2 5244 | . . . . 5 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
9 | 2 | eldm2 5244 | . . . . 5 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵) |
10 | 8, 9 | anbi12i 729 | . . . 4 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) |
11 | 7, 10 | bitri 263 | . . 3 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) |
12 | 1, 6, 11 | 3imtr4i 280 | . 2 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) → 𝑥 ∈ (dom 𝐴 ∩ dom 𝐵)) |
13 | 12 | ssriv 3572 | 1 ⊢ dom (𝐴 ∩ 𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∃wex 1695 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 〈cop 4131 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-dm 5048 |
This theorem is referenced by: rnin 5461 psssdm2 17038 hauseqcn 29269 |
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