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Theorem lubid 16813
 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 250 . . 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 3650 . . . 4 {𝑦𝐵𝑦 𝑋} ⊆ 𝐵
76a1i 11 . . 3 (𝜑 → {𝑦𝐵𝑦 𝑋} ⊆ 𝐵)
81, 2, 3, 4, 5, 7lubval 16807 . 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 16811 . . 3 ((𝜑𝑥𝐵) → ((∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ∧ ∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤)) ↔ 𝑥 = 𝑋))
119, 10riota5 6536 . 2 (𝜑 → (𝑥𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ∧ ∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤))) = 𝑋)
128, 11eqtrd 2644 1 (𝜑 → (𝑈‘{𝑦𝐵𝑦 𝑋}) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ⊆ wss 3540   class class class wbr 4583  ‘cfv 5804  ℩crio 6510  Basecbs 15695  lecple 15775  Posetcpo 16763  lubclub 16765 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-preset 16751  df-poset 16769  df-lub 16797 This theorem is referenced by:  atlatmstc  33624
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