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Mirrors > Home > MPE Home > Th. List > Mathboxes > meetdm2 | Structured version Visualization version GIF version |
Description: The meet of any two elements always exists iff all unordered pairs have GLB. (Contributed by Zhi Wang, 25-Sep-2024.) |
Ref | Expression |
---|---|
joindm2.b | ⊢ 𝐵 = (Base‘𝐾) |
joindm2.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
meetdm2.g | ⊢ 𝐺 = (glb‘𝐾) |
meetdm2.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
meetdm2 | ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | joindm2.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | meetdm2.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
3 | joindm2.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
4 | 1, 2, 3 | meetdmss 18450 | . . 3 ⊢ (𝜑 → dom ∧ ⊆ (𝐵 × 𝐵)) |
5 | eqss 4010 | . . . 4 ⊢ (dom ∧ = (𝐵 × 𝐵) ↔ (dom ∧ ⊆ (𝐵 × 𝐵) ∧ (𝐵 × 𝐵) ⊆ dom ∧ )) | |
6 | 5 | baib 535 | . . 3 ⊢ (dom ∧ ⊆ (𝐵 × 𝐵) → (dom ∧ = (𝐵 × 𝐵) ↔ (𝐵 × 𝐵) ⊆ dom ∧ )) |
7 | 4, 6 | syl 17 | . 2 ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ (𝐵 × 𝐵) ⊆ dom ∧ )) |
8 | relxp 5706 | . . 3 ⊢ Rel (𝐵 × 𝐵) | |
9 | ssrel 5794 | . . 3 ⊢ (Rel (𝐵 × 𝐵) → ((𝐵 × 𝐵) ⊆ dom ∧ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) → 〈𝑥, 𝑦〉 ∈ dom ∧ ))) | |
10 | 8, 9 | mp1i 13 | . 2 ⊢ (𝜑 → ((𝐵 × 𝐵) ⊆ dom ∧ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) → 〈𝑥, 𝑦〉 ∈ dom ∧ ))) |
11 | opelxp 5724 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) |
13 | meetdm2.g | . . . . . 6 ⊢ 𝐺 = (glb‘𝐾) | |
14 | vex 3481 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
15 | 14 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑥 ∈ V) |
16 | vex 3481 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑦 ∈ V) |
18 | 13, 2, 3, 15, 17 | meetdef 18447 | . . . . 5 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ dom ∧ ↔ {𝑥, 𝑦} ∈ dom 𝐺)) |
19 | 12, 18 | imbi12d 344 | . . . 4 ⊢ (𝜑 → ((〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) → 〈𝑥, 𝑦〉 ∈ dom ∧ ) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝐺))) |
20 | 19 | 2albidv 1920 | . . 3 ⊢ (𝜑 → (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) → 〈𝑥, 𝑦〉 ∈ dom ∧ ) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝐺))) |
21 | r2al 3192 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝐺)) | |
22 | 20, 21 | bitr4di 289 | . 2 ⊢ (𝜑 → (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) → 〈𝑥, 𝑦〉 ∈ dom ∧ ) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺)) |
23 | 7, 10, 22 | 3bitrd 305 | 1 ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1534 = wceq 1536 ∈ wcel 2105 ∀wral 3058 Vcvv 3477 ⊆ wss 3962 {cpr 4632 〈cop 4636 × cxp 5686 dom cdm 5688 Rel wrel 5693 ‘cfv 6562 Basecbs 17244 glbcglb 18367 meetcmee 18369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-oprab 7434 df-glb 18404 df-meet 18406 |
This theorem is referenced by: meetdm3 48767 toslat 48770 |
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