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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 1522 | . . 3 ⊢ (∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) → (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) | |
2 | vex 2560 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | eldm2 4533 | . . . 4 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵)) |
4 | elin 3126 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
5 | 4 | exbii 1496 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
6 | 3, 5 | bitri 173 | . . 3 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
7 | elin 3126 | . . . 4 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵)) | |
8 | 2 | eldm2 4533 | . . . . 5 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
9 | 2 | eldm2 4533 | . . . . 5 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵) |
10 | 8, 9 | anbi12i 433 | . . . 4 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) |
11 | 7, 10 | bitri 173 | . . 3 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) |
12 | 1, 6, 11 | 3imtr4i 190 | . 2 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) → 𝑥 ∈ (dom 𝐴 ∩ dom 𝐵)) |
13 | 12 | ssriv 2949 | 1 ⊢ dom (𝐴 ∩ 𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∃wex 1381 ∈ wcel 1393 ∩ cin 2916 ⊆ wss 2917 〈cop 3378 dom cdm 4345 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-dm 4355 |
This theorem is referenced by: rnin 4733 |
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