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Mirrors > Home > MPE Home > Th. List > glbeu | Structured version Visualization version GIF version |
Description: Unique existence proper of a member of the domain of the greatest lower bound function of a poset. (Contributed by NM, 7-Sep-2018.) |
Ref | Expression |
---|---|
glbval.b | ⊢ 𝐵 = (Base‘𝐾) |
glbval.l | ⊢ ≤ = (le‘𝐾) |
glbval.g | ⊢ 𝐺 = (glb‘𝐾) |
glbval.p | ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) |
glbva.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
glbval.s | ⊢ (𝜑 → 𝑆 ∈ dom 𝐺) |
Ref | Expression |
---|---|
glbeu | ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | glbval.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ dom 𝐺) | |
2 | glbval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
3 | glbval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
4 | glbval.g | . . . 4 ⊢ 𝐺 = (glb‘𝐾) | |
5 | glbval.p | . . . 4 ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) | |
6 | glbva.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
7 | 2, 3, 4, 5, 6 | glbeldm 17474 | . . 3 ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓))) |
8 | 1, 7 | mpbid 224 | . 2 ⊢ (𝜑 → (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) |
9 | 8 | simprd 488 | 1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3081 ∃!wreu 3083 ⊆ wss 3822 class class class wbr 4925 dom cdm 5403 ‘cfv 6185 Basecbs 16337 lecple 16426 glbcglb 17423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-glb 17455 |
This theorem is referenced by: glbval 17477 glbcl 17478 glbprop 17479 meeteu 17504 isglbd 17597 |
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