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Mirrors > Home > MPE Home > Th. List > meetdm | Structured version Visualization version GIF version |
Description: Domain of meet function for a poset-type structure. (Contributed by NM, 16-Sep-2018.) |
Ref | Expression |
---|---|
meetfval.u | ⊢ 𝐺 = (glb‘𝐾) |
meetfval.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
meetdm | ⊢ (𝐾 ∈ 𝑉 → dom ∧ = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meetfval.u | . . . 4 ⊢ 𝐺 = (glb‘𝐾) | |
2 | meetfval.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
3 | 1, 2 | meetfval2 17614 | . . 3 ⊢ (𝐾 ∈ 𝑉 → ∧ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}) |
4 | 3 | dmeqd 5767 | . 2 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}) |
5 | dmoprab 7244 | . . 3 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))} = {〈𝑥, 𝑦〉 ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))} | |
6 | fvex 6676 | . . . . . 6 ⊢ (𝐺‘{𝑥, 𝑦}) ∈ V | |
7 | 6 | isseti 3506 | . . . . 5 ⊢ ∃𝑧 𝑧 = (𝐺‘{𝑥, 𝑦}) |
8 | 19.42v 1945 | . . . . 5 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦})) ↔ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ ∃𝑧 𝑧 = (𝐺‘{𝑥, 𝑦}))) | |
9 | 7, 8 | mpbiran2 706 | . . . 4 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦})) ↔ {𝑥, 𝑦} ∈ dom 𝐺) |
10 | 9 | opabbii 5124 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))} = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺} |
11 | 5, 10 | eqtri 2841 | . 2 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))} = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺} |
12 | 4, 11 | syl6eq 2869 | 1 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 {cpr 4559 {copab 5119 dom cdm 5548 ‘cfv 6348 {coprab 7146 glbcglb 17541 meetcmee 17543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-oprab 7149 df-glb 17573 df-meet 17575 |
This theorem is referenced by: meetdef 17616 meetdmss 17619 |
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