| Mathbox for Zhi Wang |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > meetdm2 | Structured version Visualization version GIF version | ||
| Description: The meet of any two elements always exists iff all unordered pairs have GLB. (Contributed by Zhi Wang, 25-Sep-2024.) |
| Ref | Expression |
|---|---|
| joindm2.b | ⊢ 𝐵 = (Base‘𝐾) |
| joindm2.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
| meetdm2.g | ⊢ 𝐺 = (glb‘𝐾) |
| meetdm2.m | ⊢ ∧ = (meet‘𝐾) |
| Ref | Expression |
|---|---|
| meetdm2 | ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | joindm2.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | meetdm2.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
| 3 | joindm2.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
| 4 | 1, 2, 3 | meetdmss 18438 | . . 3 ⊢ (𝜑 → dom ∧ ⊆ (𝐵 × 𝐵)) |
| 5 | eqss 3999 | . . . 4 ⊢ (dom ∧ = (𝐵 × 𝐵) ↔ (dom ∧ ⊆ (𝐵 × 𝐵) ∧ (𝐵 × 𝐵) ⊆ dom ∧ )) | |
| 6 | 5 | baib 535 | . . 3 ⊢ (dom ∧ ⊆ (𝐵 × 𝐵) → (dom ∧ = (𝐵 × 𝐵) ↔ (𝐵 × 𝐵) ⊆ dom ∧ )) |
| 7 | 4, 6 | syl 17 | . 2 ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ (𝐵 × 𝐵) ⊆ dom ∧ )) |
| 8 | relxp 5703 | . . 3 ⊢ Rel (𝐵 × 𝐵) | |
| 9 | ssrel 5792 | . . 3 ⊢ (Rel (𝐵 × 𝐵) → ((𝐵 × 𝐵) ⊆ dom ∧ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) → 〈𝑥, 𝑦〉 ∈ dom ∧ ))) | |
| 10 | 8, 9 | mp1i 13 | . 2 ⊢ (𝜑 → ((𝐵 × 𝐵) ⊆ dom ∧ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) → 〈𝑥, 𝑦〉 ∈ dom ∧ ))) |
| 11 | opelxp 5721 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) | |
| 12 | 11 | a1i 11 | . . . . 5 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) |
| 13 | meetdm2.g | . . . . . 6 ⊢ 𝐺 = (glb‘𝐾) | |
| 14 | vex 3484 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 15 | 14 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑥 ∈ V) |
| 16 | vex 3484 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑦 ∈ V) |
| 18 | 13, 2, 3, 15, 17 | meetdef 18435 | . . . . 5 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ dom ∧ ↔ {𝑥, 𝑦} ∈ dom 𝐺)) |
| 19 | 12, 18 | imbi12d 344 | . . . 4 ⊢ (𝜑 → ((〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) → 〈𝑥, 𝑦〉 ∈ dom ∧ ) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝐺))) |
| 20 | 19 | 2albidv 1923 | . . 3 ⊢ (𝜑 → (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) → 〈𝑥, 𝑦〉 ∈ dom ∧ ) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝐺))) |
| 21 | r2al 3195 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝐺)) | |
| 22 | 20, 21 | bitr4di 289 | . 2 ⊢ (𝜑 → (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) → 〈𝑥, 𝑦〉 ∈ dom ∧ ) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺)) |
| 23 | 7, 10, 22 | 3bitrd 305 | 1 ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 {cpr 4628 〈cop 4632 × cxp 5683 dom cdm 5685 Rel wrel 5690 ‘cfv 6561 Basecbs 17247 glbcglb 18356 meetcmee 18358 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-oprab 7435 df-glb 18392 df-meet 18394 |
| This theorem is referenced by: meetdm3 48868 toslat 48871 |
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