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Theorem lubid 16930
Description: The LUB of elements less than or equal to a fixed value equals that value. (Contributed by NM, 19-Oct-2011.) (Revised by NM, 7-Sep-2018.)
Hypotheses
Ref Expression
lubid.b 𝐵 = (Base‘𝐾)
lubid.l = (le‘𝐾)
lubid.u 𝑈 = (lub‘𝐾)
lubid.k (𝜑𝐾 ∈ Poset)
lubid.x (𝜑𝑋𝐵)
Assertion
Ref Expression
lubid (𝜑 → (𝑈‘{𝑦𝐵𝑦 𝑋}) = 𝑋)
Distinct variable groups:   𝑦,   𝑦,𝐵   𝑦,𝑋
Allowed substitution hints:   𝜑(𝑦)   𝑈(𝑦)   𝐾(𝑦)

Proof of Theorem lubid
Dummy variables 𝑥 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lubid.b . . 3 𝐵 = (Base‘𝐾)
2 lubid.l . . 3 = (le‘𝐾)
3 lubid.u . . 3 𝑈 = (lub‘𝐾)
4 biid 251 . . 3 ((∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ∧ ∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤)) ↔ (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ∧ ∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤)))
5 lubid.k . . 3 (𝜑𝐾 ∈ Poset)
6 ssrab2 3672 . . . 4 {𝑦𝐵𝑦 𝑋} ⊆ 𝐵
76a1i 11 . . 3 (𝜑 → {𝑦𝐵𝑦 𝑋} ⊆ 𝐵)
81, 2, 3, 4, 5, 7lubval 16924 . 2 (𝜑 → (𝑈‘{𝑦𝐵𝑦 𝑋}) = (𝑥𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ∧ ∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤))))
9 lubid.x . . 3 (𝜑𝑋𝐵)
101, 2, 3, 5, 9lublecllem 16928 . . 3 ((𝜑𝑥𝐵) → ((∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ∧ ∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤)) ↔ 𝑥 = 𝑋))
119, 10riota5 6602 . 2 (𝜑 → (𝑥𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ∧ ∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤))) = 𝑋)
128, 11eqtrd 2655 1 (𝜑 → (𝑈‘{𝑦𝐵𝑦 𝑋}) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2908  {crab 2912  wss 3560   class class class wbr 4623  cfv 5857  crio 6575  Basecbs 15800  lecple 15888  Posetcpo 16880  lubclub 16882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-preset 16868  df-poset 16886  df-lub 16914
This theorem is referenced by:  atlatmstc  34125
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