MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lubid Structured version   Visualization version   GIF version

Theorem lubid 17592
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 264 . . 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 4007 . . . 4 {𝑦𝐵𝑦 𝑋} ⊆ 𝐵
76a1i 11 . . 3 (𝜑 → {𝑦𝐵𝑦 𝑋} ⊆ 𝐵)
81, 2, 3, 4, 5, 7lubval 17586 . 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 17590 . . 3 ((𝜑𝑥𝐵) → ((∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ∧ ∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤)) ↔ 𝑥 = 𝑋))
119, 10riota5 7122 . 2 (𝜑 → (𝑥𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ∧ ∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤))) = 𝑋)
128, 11eqtrd 2833 1 (𝜑 → (𝑈‘{𝑦𝐵𝑦 𝑋}) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  wral 3106  {crab 3110  wss 3881   class class class wbr 5030  cfv 6324  crio 7092  Basecbs 16475  lecple 16564  Posetcpo 17542  lubclub 17544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-proset 17530  df-poset 17548  df-lub 17576
This theorem is referenced by:  atlatmstc  36612
  Copyright terms: Public domain W3C validator