Theorem lubelss 18444

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

Hypotheses

Ref	Expression
lubs.b	⊢ 𝐵 = (Base‘𝐾)
lubs.l	⊢ ≤ = (le‘𝐾)
lubs.u	⊢ 𝑈 = (lub‘𝐾)
lubs.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
lubs.s	⊢ (𝜑 → 𝑆 ∈ dom 𝑈)

Assertion

Ref	Expression
lubelss	⊢ (𝜑 → 𝑆 ⊆ 𝐵)

Proof of Theorem lubelss

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

Step	Hyp	Ref	Expression
1		lubs.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
2		lubs.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3		lubs.l	. . . 4 ⊢ ≤ = (le‘𝐾)
4		lubs.u	. . . 4 ⊢ 𝑈 = (lub‘𝐾)
5		biid 264	. . . 4 ⊢ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
6		lubs.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
7	2, 3, 4, 5, 6	lubeldm 18443	. . 3 ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))))
8	1, 7	mpbid 235	. 2 ⊢ (𝜑 → (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
9	8	simpld 500	1 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃!wreu 3365   ⊆ wss 3902   class class class wbr 5107  dom cdm 5659  ‘cfv 6537  Basecbs 17305  lecple 17353  lubclub 18401

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  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 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-lub 18436

This theorem is used by:  lubcl  18447  lubprop  18448  joinfval  18463  joindmss  18469  lubsscl  49888