Theorem lubsscl 49586

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 2764	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2764	. . . . 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 18386	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐾))
8	1, 7	sstrd 3948	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐾))
9		lubsscl.x	. . . . 5 ⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝑇)
10	8, 9	sseldd 3939	. . . 4 ⊢ (𝜑 → (𝑈‘𝑆) ∈ (Base‘𝐾))
11	5	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝐾 ∈ Poset)
12	6	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑆 ∈ dom 𝑈)
13	1	sselda 3938	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑆)
14	2, 3, 4, 11, 12, 13	luble 18391	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦(le‘𝐾)(𝑈‘𝑆))
15	14	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)(𝑈‘𝑆))
16		breq1 5105	. . . . . . 7 ⊢ (𝑦 = (𝑈‘𝑆) → (𝑦(le‘𝐾)𝑧 ↔ (𝑈‘𝑆)(le‘𝐾)𝑧))
17		simp3 1152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧) → ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧)
18	9	3ad2ant1 1147	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧) → (𝑈‘𝑆) ∈ 𝑇)
19	16, 17, 18	rspcdva 3584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧) → (𝑈‘𝑆)(le‘𝐾)𝑧)
20	19	3expia 1135	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾)) → (∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → (𝑈‘𝑆)(le‘𝐾)𝑧))
21	20	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → (𝑈‘𝑆)(le‘𝐾)𝑧))
22		breq2 5106	. . . . . . 7 ⊢ (𝑥 = (𝑈‘𝑆) → (𝑦(le‘𝐾)𝑥 ↔ 𝑦(le‘𝐾)(𝑈‘𝑆)))
23	22	ralbidv 3187	. . . . . 6 ⊢ (𝑥 = (𝑈‘𝑆) → (∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑥 ↔ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)(𝑈‘𝑆)))
24		breq1 5105	. . . . . . . 8 ⊢ (𝑥 = (𝑈‘𝑆) → (𝑥(le‘𝐾)𝑧 ↔ (𝑈‘𝑆)(le‘𝐾)𝑧))
25	24	imbi2d 342	. . . . . . 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 3187	. . . . . 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 641	. . . . 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 3583	. . . 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 847	. . 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 263	. . . 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 49582	. . 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 723	. 2 ⊢ (𝜑 → 𝑇 ∈ dom 𝑈)
33	3, 2, 4, 5, 8, 10, 14, 19	poslubd 18445	. 2 ⊢ (𝜑 → (𝑈‘𝑇) = (𝑈‘𝑆))
34	32, 33	jca 519	1 ⊢ (𝜑 → (𝑇 ∈ dom 𝑈 ∧ (𝑈‘𝑇) = (𝑈‘𝑆)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088   ⊆ wss 3906   class class class wbr 5102  dom cdm 5649  ‘cfv 6523  Basecbs 17247  lecple 17295  Posetcpo 18341  lubclub 18343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-proset 18328  df-poset 18347  df-lub 18378

This theorem is referenced by:  lubprlem  49588