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Mirrors > Home > MPE Home > Th. List > meetdmss | Structured version Visualization version GIF version |
Description: Subset property of domain of meet. (Contributed by NM, 12-Sep-2018.) |
Ref | Expression |
---|---|
meetdmss.b | ⊢ 𝐵 = (Base‘𝐾) |
meetdmss.j | ⊢ ∧ = (meet‘𝐾) |
meetdmss.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
Ref | Expression |
---|---|
meetdmss | ⊢ (𝜑 → dom ∧ ⊆ (𝐵 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopabv 5833 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)} | |
2 | meetdmss.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
3 | eqid 2734 | . . . . . 6 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
4 | meetdmss.j | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
5 | 3, 4 | meetdm 18446 | . . . . 5 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)}) |
6 | 2, 5 | syl 17 | . . . 4 ⊢ (𝜑 → dom ∧ = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)}) |
7 | 6 | releqd 5790 | . . 3 ⊢ (𝜑 → (Rel dom ∧ ↔ Rel {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)})) |
8 | 1, 7 | mpbiri 258 | . 2 ⊢ (𝜑 → Rel dom ∧ ) |
9 | vex 3481 | . . . . 5 ⊢ 𝑥 ∈ V | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑥 ∈ V) |
11 | vex 3481 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑦 ∈ V) |
13 | 3, 4, 2, 10, 12 | meetdef 18447 | . . 3 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ dom ∧ ↔ {𝑥, 𝑦} ∈ dom (glb‘𝐾))) |
14 | meetdmss.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
15 | eqid 2734 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
16 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → 𝐾 ∈ 𝑉) |
17 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → {𝑥, 𝑦} ∈ dom (glb‘𝐾)) | |
18 | 14, 15, 3, 16, 17 | glbelss 18424 | . . . . 5 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵) |
19 | 18 | ex 412 | . . . 4 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵)) |
20 | 9, 11 | prss 4824 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵) |
21 | opelxpi 5725 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵)) | |
22 | 20, 21 | sylbir 235 | . . . 4 ⊢ ({𝑥, 𝑦} ⊆ 𝐵 → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵)) |
23 | 19, 22 | syl6 35 | . . 3 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵))) |
24 | 13, 23 | sylbid 240 | . 2 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ dom ∧ → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵))) |
25 | 8, 24 | relssdv 5800 | 1 ⊢ (𝜑 → dom ∧ ⊆ (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ⊆ wss 3962 {cpr 4632 〈cop 4636 {copab 5209 × cxp 5686 dom cdm 5688 Rel wrel 5693 ‘cfv 6562 Basecbs 17244 lecple 17304 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: clatl 18565 meetdm2 48766 |
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