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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 | relopab 5696 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)} | |
2 | meetdmss.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
3 | eqid 2821 | . . . . . 6 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
4 | meetdmss.j | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
5 | 3, 4 | meetdm 17627 | . . . . 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 5653 | . . 3 ⊢ (𝜑 → (Rel dom ∧ ↔ Rel {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)})) |
8 | 1, 7 | mpbiri 260 | . 2 ⊢ (𝜑 → Rel dom ∧ ) |
9 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑥 ∈ V) |
11 | vex 3497 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑦 ∈ V) |
13 | 3, 4, 2, 10, 12 | meetdef 17628 | . . 3 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ dom ∧ ↔ {𝑥, 𝑦} ∈ dom (glb‘𝐾))) |
14 | meetdmss.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
15 | eqid 2821 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
16 | 2 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → 𝐾 ∈ 𝑉) |
17 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → {𝑥, 𝑦} ∈ dom (glb‘𝐾)) | |
18 | 14, 15, 3, 16, 17 | glbelss 17605 | . . . . 5 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵) |
19 | 18 | ex 415 | . . . 4 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵)) |
20 | 9, 11 | prss 4753 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵) |
21 | opelxpi 5592 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵)) | |
22 | 20, 21 | sylbir 237 | . . . 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 242 | . 2 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ dom ∧ → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵))) |
25 | 8, 24 | relssdv 5661 | 1 ⊢ (𝜑 → dom ∧ ⊆ (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 {cpr 4569 〈cop 4573 {copab 5128 × cxp 5553 dom cdm 5555 Rel wrel 5560 ‘cfv 6355 Basecbs 16483 lecple 16572 glbcglb 17553 meetcmee 17555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-oprab 7160 df-glb 17585 df-meet 17587 |
This theorem is referenced by: clatl 17726 |
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