Theorem clatlubcl2 18562

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 6921	. . . . 5 ⊢ 𝐵 ∈ V
3	2	elpw2 5340	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵)
4	3	biimpri 228	. . 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 2735	. . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾)
8	1, 6, 7	isclat 18558	. . . 4 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom (glb‘𝐾) = 𝒫 𝐵)))
9		simprl 771	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom (glb‘𝐾) = 𝒫 𝐵)) → dom 𝑈 = 𝒫 𝐵)
10	8, 9	sylbi 217	. . 3 ⊢ (𝐾 ∈ CLat → dom 𝑈 = 𝒫 𝐵)
11	10	adantr 480	. 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → dom 𝑈 = 𝒫 𝐵)
12	5, 11	eleqtrrd 2842	1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ⊆ wss 3963  𝒫 cpw 4605  dom cdm 5689  ‘cfv 6563  Basecbs 17245  Posetcpo 18365  lubclub 18367  glbcglb 18368  CLatccla 18556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571  df-clat 18557

This theorem is referenced by:  lublem  18568