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Mirrors > Home > MPE Home > Th. List > joindmss | Structured version Visualization version GIF version |
Description: Subset property of domain of join. (Contributed by NM, 12-Sep-2018.) |
Ref | Expression |
---|---|
joindmss.b | ⊢ 𝐵 = (Base‘𝐾) |
joindmss.j | ⊢ ∨ = (join‘𝐾) |
joindmss.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
Ref | Expression |
---|---|
joindmss | ⊢ (𝜑 → dom ∨ ⊆ (𝐵 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopab 5169 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)} | |
2 | joindmss.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
3 | eqid 2610 | . . . . . 6 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
4 | joindmss.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
5 | 3, 4 | joindm 16826 | . . . . 5 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)}) |
6 | 2, 5 | syl 17 | . . . 4 ⊢ (𝜑 → dom ∨ = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)}) |
7 | 6 | releqd 5126 | . . 3 ⊢ (𝜑 → (Rel dom ∨ ↔ Rel {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)})) |
8 | 1, 7 | mpbiri 247 | . 2 ⊢ (𝜑 → Rel dom ∨ ) |
9 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑥 ∈ V) |
11 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑦 ∈ V) |
13 | 3, 4, 2, 10, 12 | joindef 16827 | . . 3 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ dom ∨ ↔ {𝑥, 𝑦} ∈ dom (lub‘𝐾))) |
14 | joindmss.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
15 | eqid 2610 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
16 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → 𝐾 ∈ 𝑉) |
17 | simpr 476 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾)) | |
18 | 14, 15, 3, 16, 17 | lubelss 16805 | . . . . 5 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵) |
19 | 18 | ex 449 | . . . 4 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵)) |
20 | 9, 11 | prss 4291 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵) |
21 | opelxpi 5072 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵)) | |
22 | 20, 21 | sylbir 224 | . . . 4 ⊢ ({𝑥, 𝑦} ⊆ 𝐵 → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵)) |
23 | 19, 22 | syl6 34 | . . 3 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵))) |
24 | 13, 23 | sylbid 229 | . 2 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ dom ∨ → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵))) |
25 | 8, 24 | relssdv 5135 | 1 ⊢ (𝜑 → dom ∨ ⊆ (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 {cpr 4127 〈cop 4131 {copab 4642 × cxp 5036 dom cdm 5038 Rel wrel 5043 ‘cfv 5804 Basecbs 15695 lecple 15775 lubclub 16765 joincjn 16767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-oprab 6553 df-lub 16797 df-join 16799 |
This theorem is referenced by: clatl 16939 |
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