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Mirrors > Home > MPE Home > Th. List > joindm | Structured version Visualization version GIF version |
Description: Domain of join function for a poset-type structure. (Contributed by NM, 16-Sep-2018.) |
Ref | Expression |
---|---|
joinfval.u | ⊢ 𝑈 = (lub‘𝐾) |
joinfval.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
joindm | ⊢ (𝐾 ∈ 𝑉 → dom ∨ = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | joinfval.u | . . . 4 ⊢ 𝑈 = (lub‘𝐾) | |
2 | joinfval.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
3 | 1, 2 | joinfval2 18090 | . . 3 ⊢ (𝐾 ∈ 𝑉 → ∨ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}) |
4 | 3 | dmeqd 5813 | . 2 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}) |
5 | dmoprab 7370 | . . 3 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {〈𝑥, 𝑦〉 ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} | |
6 | fvex 6784 | . . . . . 6 ⊢ (𝑈‘{𝑥, 𝑦}) ∈ V | |
7 | 6 | isseti 3446 | . . . . 5 ⊢ ∃𝑧 𝑧 = (𝑈‘{𝑥, 𝑦}) |
8 | 19.42v 1961 | . . . . 5 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ ∃𝑧 𝑧 = (𝑈‘{𝑥, 𝑦}))) | |
9 | 7, 8 | mpbiran2 707 | . . . 4 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ {𝑥, 𝑦} ∈ dom 𝑈) |
10 | 9 | opabbii 5146 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈} |
11 | 5, 10 | eqtri 2768 | . 2 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈} |
12 | 4, 11 | eqtrdi 2796 | 1 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∃wex 1786 ∈ wcel 2110 {cpr 4569 {copab 5141 dom cdm 5590 ‘cfv 6432 {coprab 7272 lubclub 18025 joincjn 18027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-oprab 7275 df-lub 18062 df-join 18064 |
This theorem is referenced by: joindef 18092 joindmss 18095 |
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