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Mirrors > Home > MPE Home > Th. List > Mathboxes > joindm2 | Structured version Visualization version GIF version |
Description: The join of any two elements always exists iff all unordered pairs have LUB. (Contributed by Zhi Wang, 25-Sep-2024.) |
Ref | Expression |
---|---|
joindm2.b | ⊢ 𝐵 = (Base‘𝐾) |
joindm2.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
joindm2.u | ⊢ 𝑈 = (lub‘𝐾) |
joindm2.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
joindm2 | ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | joindm2.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | joindm2.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
3 | joindm2.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
4 | 1, 2, 3 | joindmss 17839 | . . 3 ⊢ (𝜑 → dom ∨ ⊆ (𝐵 × 𝐵)) |
5 | eqss 3902 | . . . 4 ⊢ (dom ∨ = (𝐵 × 𝐵) ↔ (dom ∨ ⊆ (𝐵 × 𝐵) ∧ (𝐵 × 𝐵) ⊆ dom ∨ )) | |
6 | 5 | baib 539 | . . 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 5554 | . . 3 ⊢ Rel (𝐵 × 𝐵) | |
9 | ssrel 5639 | . . 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 5572 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) |
13 | joindm2.u | . . . . . 6 ⊢ 𝑈 = (lub‘𝐾) | |
14 | vex 3402 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
15 | 14 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑥 ∈ V) |
16 | vex 3402 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑦 ∈ V) |
18 | 13, 2, 3, 15, 17 | joindef 17836 | . . . . 5 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ dom ∨ ↔ {𝑥, 𝑦} ∈ dom 𝑈)) |
19 | 12, 18 | imbi12d 348 | . . . 4 ⊢ (𝜑 → ((〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) → 〈𝑥, 𝑦〉 ∈ dom ∨ ) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝑈))) |
20 | 19 | 2albidv 1931 | . . 3 ⊢ (𝜑 → (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) → 〈𝑥, 𝑦〉 ∈ dom ∨ ) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝑈))) |
21 | r2al 3112 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝑈)) | |
22 | 20, 21 | bitr4di 292 | . 2 ⊢ (𝜑 → (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐵) → 〈𝑥, 𝑦〉 ∈ dom ∨ ) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈)) |
23 | 7, 10, 22 | 3bitrd 308 | 1 ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 ∈ wcel 2112 ∀wral 3051 Vcvv 3398 ⊆ wss 3853 {cpr 4529 〈cop 4533 × cxp 5534 dom cdm 5536 Rel wrel 5541 ‘cfv 6358 Basecbs 16666 lubclub 17770 joincjn 17772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-oprab 7195 df-lub 17806 df-join 17808 |
This theorem is referenced by: joindm3 45879 toslat 45884 |
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