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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 5247 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)} | |
2 | joindmss.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
3 | eqid 2622 | . . . . . 6 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
4 | joindmss.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
5 | 3, 4 | joindm 17003 | . . . . 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 5203 | . . 3 ⊢ (𝜑 → (Rel dom ∨ ↔ Rel {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)})) |
8 | 1, 7 | mpbiri 248 | . 2 ⊢ (𝜑 → Rel dom ∨ ) |
9 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑥 ∈ V) |
11 | vex 3203 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑦 ∈ V) |
13 | 3, 4, 2, 10, 12 | joindef 17004 | . . 3 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ dom ∨ ↔ {𝑥, 𝑦} ∈ dom (lub‘𝐾))) |
14 | joindmss.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
15 | eqid 2622 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
16 | 2 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → 𝐾 ∈ 𝑉) |
17 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾)) | |
18 | 14, 15, 3, 16, 17 | lubelss 16982 | . . . . 5 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵) |
19 | 18 | ex 450 | . . . 4 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵)) |
20 | 9, 11 | prss 4351 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵) |
21 | opelxpi 5148 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵)) | |
22 | 20, 21 | sylbir 225 | . . . 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 230 | . 2 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ dom ∨ → 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵))) |
25 | 8, 24 | relssdv 5212 | 1 ⊢ (𝜑 → dom ∨ ⊆ (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 {cpr 4179 〈cop 4183 {copab 4712 × cxp 5112 dom cdm 5114 Rel wrel 5119 ‘cfv 5888 Basecbs 15857 lecple 15948 lubclub 16942 joincjn 16944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-oprab 6654 df-lub 16974 df-join 16976 |
This theorem is referenced by: clatl 17116 |
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