Theorem clatlubcl2 18459

Description: Any subset of the base set has an LUB in a complete lattice. (Contributed by NM, 13-Sep-2018.)

Hypotheses

Ref	Expression
clatlubcl.b	⊢ 𝐵 = (Base‘𝐾)
clatlubcl.u	⊢ 𝑈 = (lub‘𝐾)

Assertion

Ref	Expression
clatlubcl2	⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝑈)

Proof of Theorem clatlubcl2

Step	Hyp	Ref	Expression
1		clatlubcl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
2	1	fvexi 6895	. . . . 5 ⊢ 𝐵 ∈ V
3	2	elpw2 5335	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵)
4	3	biimpri 227	. . 3 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ∈ 𝒫 𝐵)
5	4	adantl 481	. 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ 𝒫 𝐵)
6		clatlubcl.u	. . . . 5 ⊢ 𝑈 = (lub‘𝐾)
7		eqid 2724	. . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾)
8	1, 6, 7	isclat 18455	. . . 4 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom (glb‘𝐾) = 𝒫 𝐵)))
9		simprl 768	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom (glb‘𝐾) = 𝒫 𝐵)) → dom 𝑈 = 𝒫 𝐵)
10	8, 9	sylbi 216	. . 3 ⊢ (𝐾 ∈ CLat → dom 𝑈 = 𝒫 𝐵)
11	10	adantr 480	. 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → dom 𝑈 = 𝒫 𝐵)
12	5, 11	eleqtrrd 2828	1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ⊆ wss 3940  𝒫 cpw 4594  dom cdm 5666  ‘cfv 6533  Basecbs 17143  Posetcpo 18262  lubclub 18264  glbcglb 18265  CLatccla 18453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-dm 5676  df-iota 6485  df-fv 6541  df-clat 18454

This theorem is referenced by:  lublem  18465