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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 | relopabv 5825 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)} | |
2 | joindmss.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
3 | eqid 2727 | . . . . . 6 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
4 | joindmss.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
5 | 3, 4 | joindm 18372 | . . . . 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 5782 | . . 3 ⊢ (𝜑 → (Rel dom ∨ ↔ Rel {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)})) |
8 | 1, 7 | mpbiri 257 | . 2 ⊢ (𝜑 → Rel dom ∨ ) |
9 | vex 3475 | . . . . 5 ⊢ 𝑥 ∈ V | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑥 ∈ V) |
11 | vex 3475 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑦 ∈ V) |
13 | 3, 4, 2, 10, 12 | joindef 18373 | . . 3 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ dom ∨ ↔ {𝑥, 𝑦} ∈ dom (lub‘𝐾))) |
14 | joindmss.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
15 | eqid 2727 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
16 | 2 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → 𝐾 ∈ 𝑉) |
17 | simpr 483 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾)) | |
18 | 14, 15, 3, 16, 17 | lubelss 18351 | . . . . 5 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵) |
19 | 18 | ex 411 | . . . 4 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵)) |
20 | 9, 11 | prss 4826 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵) |
21 | opelxpi 5717 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵)) | |
22 | 20, 21 | sylbir 234 | . . . 4 ⊢ ({𝑥, 𝑦} ⊆ 𝐵 → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵)) |
23 | 19, 22 | syl6 35 | . . 3 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵))) |
24 | 13, 23 | sylbid 239 | . 2 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ dom ∨ → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵))) |
25 | 8, 24 | relssdv 5792 | 1 ⊢ (𝜑 → dom ∨ ⊆ (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3471 ⊆ wss 3947 {cpr 4632 〈cop 4636 {copab 5212 × cxp 5678 dom cdm 5680 Rel wrel 5685 ‘cfv 6551 Basecbs 17185 lecple 17245 lubclub 18306 joincjn 18308 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-oprab 7428 df-lub 18343 df-join 18345 |
This theorem is referenced by: clatl 18505 joindm2 48038 |
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