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Mirrors > Home > MPE Home > Th. List > lubcl | Structured version Visualization version GIF version |
Description: The least upper bound function value belongs to the base set. (Contributed by NM, 7-Sep-2018.) |
Ref | Expression |
---|---|
lubcl.b | ⊢ 𝐵 = (Base‘𝐾) |
lubcl.u | ⊢ 𝑈 = (lub‘𝐾) |
lubcl.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
lubcl.s | ⊢ (𝜑 → 𝑆 ∈ dom 𝑈) |
Ref | Expression |
---|---|
lubcl | ⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lubcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2821 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | lubcl.u | . . 3 ⊢ 𝑈 = (lub‘𝐾) | |
4 | biid 263 | . . 3 ⊢ ((∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))) | |
5 | lubcl.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
6 | lubcl.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈) | |
7 | 1, 2, 3, 5, 6 | lubelss 17592 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
8 | 1, 2, 3, 4, 5, 7 | lubval 17594 | . 2 ⊢ (𝜑 → (𝑈‘𝑆) = (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) |
9 | 1, 2, 3, 4, 5, 6 | lubeu 17593 | . . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))) |
10 | riotacl 7131 | . . 3 ⊢ (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))) ∈ 𝐵) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))) ∈ 𝐵) |
12 | 8, 11 | eqeltrd 2913 | 1 ⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃!wreu 3140 class class class wbr 5066 dom cdm 5555 ‘cfv 6355 ℩crio 7113 Basecbs 16483 lecple 16572 lubclub 17552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-lub 17584 |
This theorem is referenced by: lubprop 17596 joincl 17616 clatlem 17721 op1cl 36336 |
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