Theorem glbelss 18417

Description: A member of the domain of the greatest lower bound function is a subset of the base set. (Contributed by NM, 7-Sep-2018.)

Hypotheses

Ref	Expression
glbs.b	⊢ 𝐵 = (Base‘𝐾)
glbs.l	⊢ ≤ = (le‘𝐾)
glbs.g	⊢ 𝐺 = (glb‘𝐾)
glbs.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
glbs.s	⊢ (𝜑 → 𝑆 ∈ dom 𝐺)

Assertion

Ref	Expression
glbelss	⊢ (𝜑 → 𝑆 ⊆ 𝐵)

Proof of Theorem glbelss

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

Step	Hyp	Ref	Expression
1		glbs.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ dom 𝐺)
2		glbs.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3		glbs.l	. . . 4 ⊢ ≤ = (le‘𝐾)
4		glbs.g	. . . 4 ⊢ 𝐺 = (glb‘𝐾)
5		biid 264	. . . 4 ⊢ ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
6		glbs.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
7	2, 3, 4, 5, 6	glbeldm 18416	. . 3 ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))))
8	1, 7	mpbid 235	. 2 ⊢ (𝜑 → (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
9	8	simpld 499	1 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃!wreu 3374   ⊆ wss 3913   class class class wbr 5110  dom cdm 5659  ‘cfv 6534  Basecbs 17265  lecple 17313  glbcglb 18362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-glb 18397

This theorem is referenced by:  glbcl  18420  glbprop  18421  meetfval  18437  meetdmss  18443  glbsscl  49619