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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 2818 | . . 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 17603 | . 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 17602 | . . . 4 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∨ ↔ {𝑋, 𝑌} ∈ dom (lub‘𝐾))) |
10 | 8, 9 | mpbid 233 | . . 3 ⊢ (𝜑 → {𝑋, 𝑌} ∈ dom (lub‘𝐾)) |
11 | 7, 1, 3, 10 | lubcl 17583 | . 2 ⊢ (𝜑 → ((lub‘𝐾)‘{𝑋, 𝑌}) ∈ 𝐵) |
12 | 6, 11 | eqeltrd 2910 | 1 ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {cpr 4559 〈cop 4563 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 lubclub 17540 joincjn 17542 |
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-ov 7148 df-oprab 7149 df-lub 17572 df-join 17574 |
This theorem is referenced by: joinle 17612 latlem 17647 |
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