Theorem luble 17591

Description: The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)

Hypotheses

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

Assertion

Ref	Expression
luble	⊢ (𝜑 → 𝑋 ≤ (𝑈‘𝑆))

Proof of Theorem luble

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

Step	Hyp	Ref	Expression
1		breq1 5061	. 2 ⊢ (𝑦 = 𝑋 → (𝑦 ≤ (𝑈‘𝑆) ↔ 𝑋 ≤ (𝑈‘𝑆)))
2		lubprop.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3		lubprop.l	. . . 4 ⊢ ≤ = (le‘𝐾)
4		lubprop.u	. . . 4 ⊢ 𝑈 = (lub‘𝐾)
5		lubprop.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
6		lubprop.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
7	2, 3, 4, 5, 6	lubprop 17590	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧)))
8	7	simpld 497	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆))
9		luble.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
10	1, 8, 9	rspcdva 3624	1 ⊢ (𝜑 → 𝑋 ≤ (𝑈‘𝑆))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  ∀wral 3138   class class class wbr 5058  dom cdm 5549  ‘cfv 6349  Basecbs 16477  lecple 16566  lubclub 17546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-lub 17578

This theorem is referenced by:  ple1  17648