Theorem lubelss 18277

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 261	. . . 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 18276	. . 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 232	. 2 ⊢ (𝜑 → (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
9	8	simpld 494	1 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3050  ∃!wreu 3347   ⊆ wss 3900   class class class wbr 5097  dom cdm 5623  ‘cfv 6491  Basecbs 17138  lecple 17186  lubclub 18234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-lub 18269

This theorem is referenced by:  lubcl  18280  lubprop  18281  joinfval  18296  joindmss  18302  lubsscl  49242