Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 1883 | . . 3 ⊢ (∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) → (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) | |
2 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | eldm2 5769 | . . . 4 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵)) |
4 | elin 4168 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
5 | 4 | exbii 1844 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
6 | 3, 5 | bitri 277 | . . 3 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
7 | elin 4168 | . . . 4 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵)) | |
8 | 2 | eldm2 5769 | . . . . 5 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
9 | 2 | eldm2 5769 | . . . . 5 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵) |
10 | 8, 9 | anbi12i 628 | . . . 4 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) |
11 | 7, 10 | bitri 277 | . . 3 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) |
12 | 1, 6, 11 | 3imtr4i 294 | . 2 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) → 𝑥 ∈ (dom 𝐴 ∩ dom 𝐵)) |
13 | 12 | ssriv 3970 | 1 ⊢ dom (𝐴 ∩ 𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∃wex 1776 ∈ wcel 2110 ∩ cin 3934 ⊆ wss 3935 〈cop 4572 dom cdm 5554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-dm 5564 |
This theorem is referenced by: rnin 6004 psssdm2 17824 hauseqcn 31138 |
Copyright terms: Public domain | W3C validator |