Theorem clatglbcl2 18551

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

Hypotheses

Ref	Expression
clatglbcl.b	⊢ 𝐵 = (Base‘𝐾)
clatglbcl.g	⊢ 𝐺 = (glb‘𝐾)

Assertion

Ref	Expression
clatglbcl2	⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺)

Proof of Theorem clatglbcl2

Step	Hyp	Ref	Expression
1		clatglbcl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
2	1	fvexi 6920	. . . . 5 ⊢ 𝐵 ∈ V
3	2	elpw2 5334	. . . 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		eqid 2737	. . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾)
7		clatglbcl.g	. . . . 5 ⊢ 𝐺 = (glb‘𝐾)
8	1, 6, 7	isclat 18545	. . . 4 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
9		simprr 773	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)) → dom 𝐺 = 𝒫 𝐵)
10	8, 9	sylbi 217	. . 3 ⊢ (𝐾 ∈ CLat → dom 𝐺 = 𝒫 𝐵)
11	10	adantr 480	. 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → dom 𝐺 = 𝒫 𝐵)
12	5, 11	eleqtrrd 2844	1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  𝒫 cpw 4600  dom cdm 5685  ‘cfv 6561  Basecbs 17247  Posetcpo 18353  lubclub 18355  glbcglb 18356  CLatccla 18543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-dm 5695  df-iota 6514  df-fv 6569  df-clat 18544

This theorem is referenced by:  isglbd  18554  clatglb  18561  clatglble  18562  glbconN  39378