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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 16825 | . . 3 ⊢ (𝐾 ∈ 𝑉 → ∨ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}) |
4 | 3 | dmeqd 5248 | . 2 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}) |
5 | dmoprab 6639 | . . 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 6113 | . . . . . 6 ⊢ (𝑈‘{𝑥, 𝑦}) ∈ V | |
7 | 6 | isseti 3182 | . . . . 5 ⊢ ∃𝑧 𝑧 = (𝑈‘{𝑥, 𝑦}) |
8 | 19.42v 1905 | . . . . 5 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ ∃𝑧 𝑧 = (𝑈‘{𝑥, 𝑦}))) | |
9 | 7, 8 | mpbiran2 956 | . . . 4 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ {𝑥, 𝑦} ∈ dom 𝑈) |
10 | 9 | opabbii 4649 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈} |
11 | 5, 10 | eqtri 2632 | . 2 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈} |
12 | 4, 11 | syl6eq 2660 | 1 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cpr 4127 {copab 4642 dom cdm 5038 ‘cfv 5804 {coprab 6550 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: joindef 16827 joindmss 16830 |
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