Theorem glbelss 18589

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 18588	. . 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 500	1 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃!wreu 3367   ⊆ wss 3900   class class class wbr 5108  dom cdm 5658  ‘cfv 6566  Basecbs 17431  lecple 17479  glbcglb 18534

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-glb 18569

This theorem is used by:  glbcl  18592  glbprop  18593  meetfval  18609  meetdmss  18615  glbsscl  50406