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Mirrors > Home > MPE Home > Th. List > dmin | Structured version Visualization version GIF version |
Description: The domain of an intersection is included in the intersection of the 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 1882 | . . 3 ⊢ (∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵)) | |
2 | vex 3474 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | eldm2 5899 | . . . 4 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∩ 𝐵)) |
4 | elin 3961 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∩ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) | |
5 | 4 | exbii 1843 | . . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) |
6 | 3, 5 | bitri 275 | . . 3 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) |
7 | elin 3961 | . . . 4 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵)) | |
8 | 2 | eldm2 5899 | . . . . 5 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴) |
9 | 2 | eldm2 5899 | . . . . 5 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵) |
10 | 8, 9 | anbi12i 627 | . . . 4 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵)) |
11 | 7, 10 | bitri 275 | . . 3 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵)) |
12 | 1, 6, 11 | 3imtr4i 292 | . 2 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) → 𝑥 ∈ (dom 𝐴 ∩ dom 𝐵)) |
13 | 12 | ssriv 3983 | 1 ⊢ dom (𝐴 ∩ 𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∃wex 1774 ∈ wcel 2099 ∩ cin 3944 ⊆ wss 3945 ⟨cop 4631 dom cdm 5673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-br 5144 df-dm 5683 |
This theorem is referenced by: rnin 6146 psssdm2 18567 hauseqcn 33494 |
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