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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 1894 | . . 3 ⊢ (∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) → (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) | |
2 | vex 3402 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | eldm2 5755 | . . . 4 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵)) |
4 | elin 3869 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
5 | 4 | exbii 1855 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
6 | 3, 5 | bitri 278 | . . 3 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
7 | elin 3869 | . . . 4 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵)) | |
8 | 2 | eldm2 5755 | . . . . 5 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
9 | 2 | eldm2 5755 | . . . . 5 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵) |
10 | 8, 9 | anbi12i 630 | . . . 4 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) |
11 | 7, 10 | bitri 278 | . . 3 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) |
12 | 1, 6, 11 | 3imtr4i 295 | . 2 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) → 𝑥 ∈ (dom 𝐴 ∩ dom 𝐵)) |
13 | 12 | ssriv 3891 | 1 ⊢ dom (𝐴 ∩ 𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∃wex 1787 ∈ wcel 2112 ∩ cin 3852 ⊆ wss 3853 〈cop 4533 dom cdm 5536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-dm 5546 |
This theorem is referenced by: rnin 5990 psssdm2 18041 hauseqcn 31516 |
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