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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 17606 | . . 3 ⊢ (𝐾 ∈ 𝑉 → ∨ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}) |
4 | 3 | dmeqd 5768 | . 2 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}) |
5 | dmoprab 7249 | . . 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 6677 | . . . . . 6 ⊢ (𝑈‘{𝑥, 𝑦}) ∈ V | |
7 | 6 | isseti 3508 | . . . . 5 ⊢ ∃𝑧 𝑧 = (𝑈‘{𝑥, 𝑦}) |
8 | 19.42v 1950 | . . . . 5 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ ∃𝑧 𝑧 = (𝑈‘{𝑥, 𝑦}))) | |
9 | 7, 8 | mpbiran2 708 | . . . 4 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ {𝑥, 𝑦} ∈ dom 𝑈) |
10 | 9 | opabbii 5125 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈} |
11 | 5, 10 | eqtri 2844 | . 2 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈} |
12 | 4, 11 | syl6eq 2872 | 1 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cpr 4562 {copab 5120 dom cdm 5549 ‘cfv 6349 {coprab 7151 lubclub 17546 joincjn 17548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-oprab 7154 df-lub 17578 df-join 17580 |
This theorem is referenced by: joindef 17608 joindmss 17611 |
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