![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 1519 | . . 3 ⊢ (∃y(〈x, y〉 ∈ A ∧ 〈x, y〉 ∈ B) → (∃y〈x, y〉 ∈ A ∧ ∃y〈x, y〉 ∈ B)) | |
2 | vex 2554 | . . . . 5 ⊢ x ∈ V | |
3 | 2 | eldm2 4476 | . . . 4 ⊢ (x ∈ dom (A ∩ B) ↔ ∃y〈x, y〉 ∈ (A ∩ B)) |
4 | elin 3120 | . . . . 5 ⊢ (〈x, y〉 ∈ (A ∩ B) ↔ (〈x, y〉 ∈ A ∧ 〈x, y〉 ∈ B)) | |
5 | 4 | exbii 1493 | . . . 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 3120 | . . . 4 ⊢ (x ∈ (dom A ∩ dom B) ↔ (x ∈ dom A ∧ x ∈ dom B)) | |
8 | 2 | eldm2 4476 | . . . . 5 ⊢ (x ∈ dom A ↔ ∃y〈x, y〉 ∈ A) |
9 | 2 | eldm2 4476 | . . . . 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 2943 | 1 ⊢ dom (A ∩ B) ⊆ (dom A ∩ dom B) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∃wex 1378 ∈ wcel 1390 ∩ cin 2910 ⊆ wss 2911 〈cop 3370 dom cdm 4288 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-dm 4298 |
This theorem is referenced by: rnin 4676 |
Copyright terms: Public domain | W3C validator |