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Mirrors > Home > MPE Home > Th. List > joincl | Structured version Visualization version GIF version |
Description: Closure of join of elements in the domain. (Contributed by NM, 12-Sep-2018.) |
Ref | Expression |
---|---|
joincl.b | ⊢ 𝐵 = (Base‘𝐾) |
joincl.j | ⊢ ∨ = (join‘𝐾) |
joincl.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
joincl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
joincl.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
joincl.e | ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
Ref | Expression |
---|---|
joincl | ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
2 | joincl.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | joincl.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
4 | joincl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | joincl.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | joinval 16828 | . 2 ⊢ (𝜑 → (𝑋 ∨ 𝑌) = ((lub‘𝐾)‘{𝑋, 𝑌})) |
7 | joincl.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
8 | joincl.e | . . . 4 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) | |
9 | 1, 2, 3, 4, 5 | joindef 16827 | . . . 4 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∨ ↔ {𝑋, 𝑌} ∈ dom (lub‘𝐾))) |
10 | 8, 9 | mpbid 221 | . . 3 ⊢ (𝜑 → {𝑋, 𝑌} ∈ dom (lub‘𝐾)) |
11 | 7, 1, 3, 10 | lubcl 16808 | . 2 ⊢ (𝜑 → ((lub‘𝐾)‘{𝑋, 𝑌}) ∈ 𝐵) |
12 | 6, 11 | eqeltrd 2688 | 1 ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {cpr 4127 〈cop 4131 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 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-ov 6552 df-oprab 6553 df-lub 16797 df-join 16799 |
This theorem is referenced by: joinle 16837 latlem 16872 |
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