Theorem glbelss 18113

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 260	. . . 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 18112	. . 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 231	. 2 ⊢ (𝜑 → (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
9	8	simpld 494	1 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2101  ∀wral 3059  ∃!wreu 3219   ⊆ wss 3889   class class class wbr 5077  dom cdm 5591  ‘cfv 6447  Basecbs 16940  lecple 16997  glbcglb 18056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-riota 7252  df-glb 18093

This theorem is referenced by:  glbcl  18116  glbprop  18117  meetfval  18133  meetdmss  18139  glbsscl  46295