Theorem lubsscl 46254

Description: If a subset of 𝑆 contains the LUB of 𝑆, then the two sets have the same LUB. (Contributed by Zhi Wang, 26-Sep-2024.)

Hypotheses

Ref	Expression
lubsscl.k	⊢ (𝜑 → 𝐾 ∈ Poset)
lubsscl.t	⊢ (𝜑 → 𝑇 ⊆ 𝑆)
lubsscl.u	⊢ 𝑈 = (lub‘𝐾)
lubsscl.s	⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
lubsscl.x	⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝑇)

Assertion

Ref	Expression
lubsscl	⊢ (𝜑 → (𝑇 ∈ dom 𝑈 ∧ (𝑈‘𝑇) = (𝑈‘𝑆)))

Proof of Theorem lubsscl

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

Step	Hyp	Ref	Expression
1		lubsscl.t	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑆)
2		eqid 2738	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2738	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
4		lubsscl.u	. . . . 5 ⊢ 𝑈 = (lub‘𝐾)
5		lubsscl.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Poset)
6		lubsscl.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
7	2, 3, 4, 5, 6	lubelss 18072	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐾))
8	1, 7	sstrd 3931	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐾))
9		lubsscl.x	. . . . 5 ⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝑇)
10	8, 9	sseldd 3922	. . . 4 ⊢ (𝜑 → (𝑈‘𝑆) ∈ (Base‘𝐾))
11	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝐾 ∈ Poset)
12	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑆 ∈ dom 𝑈)
13	1	sselda 3921	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑆)
14	2, 3, 4, 11, 12, 13	luble 18077	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦(le‘𝐾)(𝑈‘𝑆))
15	14	ralrimiva 3103	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)(𝑈‘𝑆))
16		breq1 5077	. . . . . . 7 ⊢ (𝑦 = (𝑈‘𝑆) → (𝑦(le‘𝐾)𝑧 ↔ (𝑈‘𝑆)(le‘𝐾)𝑧))
17		simp3 1137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧) → ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧)
18	9	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧) → (𝑈‘𝑆) ∈ 𝑇)
19	16, 17, 18	rspcdva 3562	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧) → (𝑈‘𝑆)(le‘𝐾)𝑧)
20	19	3expia 1120	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾)) → (∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → (𝑈‘𝑆)(le‘𝐾)𝑧))
21	20	ralrimiva 3103	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → (𝑈‘𝑆)(le‘𝐾)𝑧))
22		breq2 5078	. . . . . . 7 ⊢ (𝑥 = (𝑈‘𝑆) → (𝑦(le‘𝐾)𝑥 ↔ 𝑦(le‘𝐾)(𝑈‘𝑆)))
23	22	ralbidv 3112	. . . . . 6 ⊢ (𝑥 = (𝑈‘𝑆) → (∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑥 ↔ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)(𝑈‘𝑆)))
24		breq1 5077	. . . . . . . 8 ⊢ (𝑥 = (𝑈‘𝑆) → (𝑥(le‘𝐾)𝑧 ↔ (𝑈‘𝑆)(le‘𝐾)𝑧))
25	24	imbi2d 341	. . . . . . 7 ⊢ (𝑥 = (𝑈‘𝑆) → ((∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧) ↔ (∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → (𝑈‘𝑆)(le‘𝐾)𝑧)))
26	25	ralbidv 3112	. . . . . 6 ⊢ (𝑥 = (𝑈‘𝑆) → (∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧) ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → (𝑈‘𝑆)(le‘𝐾)𝑧)))
27	23, 26	anbi12d 631	. . . . 5 ⊢ (𝑥 = (𝑈‘𝑆) → ((∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)(𝑈‘𝑆) ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → (𝑈‘𝑆)(le‘𝐾)𝑧))))
28	27	rspcev 3561	. . . 4 ⊢ (((𝑈‘𝑆) ∈ (Base‘𝐾) ∧ (∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)(𝑈‘𝑆) ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → (𝑈‘𝑆)(le‘𝐾)𝑧))) → ∃𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
29	10, 15, 21, 28	syl12anc 834	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
30		biid 260	. . . 4 ⊢ ((∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
31	2, 3, 4, 30, 5	lubeldm2 46250	. . 3 ⊢ (𝜑 → (𝑇 ∈ dom 𝑈 ↔ (𝑇 ⊆ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))))
32	8, 29, 31	mpbir2and 710	. 2 ⊢ (𝜑 → 𝑇 ∈ dom 𝑈)
33	3, 2, 4, 5, 8, 10, 14, 19	poslubd 18131	. 2 ⊢ (𝜑 → (𝑈‘𝑇) = (𝑈‘𝑆))
34	32, 33	jca 512	1 ⊢ (𝜑 → (𝑇 ∈ dom 𝑈 ∧ (𝑈‘𝑇) = (𝑈‘𝑆)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887   class class class wbr 5074  dom cdm 5589  ‘cfv 6433  Basecbs 16912  lecple 16969  Posetcpo 18025  lubclub 18027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-proset 18013  df-poset 18031  df-lub 18064

This theorem is referenced by:  lubprlem  46256