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Mirrors > Home > MPE Home > Th. List > glbelss | Structured version Visualization version GIF version |
Description: A member of the domain of the greatest lower bound function is a subset of the base set. (Contributed by NM, 7-Sep-2018.) |
Ref | Expression |
---|---|
glbs.b | ⊢ 𝐵 = (Base‘𝐾) |
glbs.l | ⊢ ≤ = (le‘𝐾) |
glbs.g | ⊢ 𝐺 = (glb‘𝐾) |
glbs.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
glbs.s | ⊢ (𝜑 → 𝑆 ∈ dom 𝐺) |
Ref | Expression |
---|---|
glbelss | ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | glbs.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ dom 𝐺) | |
2 | glbs.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
3 | glbs.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
4 | glbs.g | . . . 4 ⊢ 𝐺 = (glb‘𝐾) | |
5 | biid 262 | . . . 4 ⊢ ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) | |
6 | glbs.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
7 | 2, 3, 4, 5, 6 | glbeldm 17592 | . . 3 ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))) |
8 | 1, 7 | mpbid 233 | . 2 ⊢ (𝜑 → (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))) |
9 | 8 | simpld 495 | 1 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃!wreu 3137 ⊆ wss 3933 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 Basecbs 16471 lecple 16560 glbcglb 17541 |
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 |
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-glb 17573 |
This theorem is referenced by: glbcl 17596 glbprop 17597 meetfval 17613 meetdmss 17619 |
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