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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 5699 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)} | |
2 | meetdmss.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
3 | eqid 2824 | . . . . . 6 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
4 | meetdmss.j | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
5 | 3, 4 | meetdm 17630 | . . . . 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 5656 | . . 3 ⊢ (𝜑 → (Rel dom ∧ ↔ Rel {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)})) |
8 | 1, 7 | mpbiri 260 | . 2 ⊢ (𝜑 → Rel dom ∧ ) |
9 | vex 3500 | . . . . 5 ⊢ 𝑥 ∈ V | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑥 ∈ V) |
11 | vex 3500 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑦 ∈ V) |
13 | 3, 4, 2, 10, 12 | meetdef 17631 | . . 3 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ dom ∧ ↔ {𝑥, 𝑦} ∈ dom (glb‘𝐾))) |
14 | meetdmss.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
15 | eqid 2824 | . . . . . 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 17608 | . . . . 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 4756 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵) |
21 | opelxpi 5595 | . . . . 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 5664 | 1 ⊢ (𝜑 → dom ∧ ⊆ (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 {cpr 4572 〈cop 4576 {copab 5131 × cxp 5556 dom cdm 5558 Rel wrel 5563 ‘cfv 6358 Basecbs 16486 lecple 16575 glbcglb 17556 meetcmee 17558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-oprab 7163 df-glb 17588 df-meet 17590 |
This theorem is referenced by: clatl 17729 |
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