Theorem lublecl 16812

Description: The set of all elements less than a given element has an LUB. (Contributed by NM, 8-Sep-2018.)

Hypotheses

Ref	Expression
lublecl.b	⊢ 𝐵 = (Base‘𝐾)
lublecl.l	⊢ ≤ = (le‘𝐾)
lublecl.u	⊢ 𝑈 = (lub‘𝐾)
lublecl.k	⊢ (𝜑 → 𝐾 ∈ Poset)
lublecl.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
lublecl	⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋} ∈ dom 𝑈)

Distinct variable groups:   𝑦, ≤   𝑦,𝐵   𝑦,𝑋

Allowed substitution hints:   𝜑(𝑦)   𝑈(𝑦)   𝐾(𝑦)

Proof of Theorem lublecl

Dummy variables 𝑥 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3650	. . 3 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵
2	1	a1i 11	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵)
3		lublecl.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
4		lublecl.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
5		lublecl.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
6		lublecl.u	. . . . 5 ⊢ 𝑈 = (lub‘𝐾)
7		lublecl.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Poset)
8	4, 5, 6, 7, 3	lublecllem 16811	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)) ↔ 𝑥 = 𝑋))
9	8	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)) ↔ 𝑥 = 𝑋))
10		reu6i 3364	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)) ↔ 𝑥 = 𝑋)) → ∃!𝑥 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)))
11	3, 9, 10	syl2anc 691	. 2 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)))
12		biid 250	. . 3 ⊢ ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)) ↔ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)))
13	4, 5, 6, 12, 7	lubeldm 16804	. 2 ⊢ (𝜑 → ({𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋} ∈ dom 𝑈 ↔ ({𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)))))
14	2, 11, 13	mpbir2and 959	1 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋} ∈ dom 𝑈)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃!wreu 2898  {crab 2900   ⊆ wss 3540   class class class wbr 4583  dom cdm 5038  ‘cfv 5804  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-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: (None)