Theorem lubsscl 49766

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 2762	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2762	. . . . 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 18414	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐾))
8	1, 7	sstrd 3946	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐾))
9		lubsscl.x	. . . . 5 ⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝑇)
10	8, 9	sseldd 3937	. . . 4 ⊢ (𝜑 → (𝑈‘𝑆) ∈ (Base‘𝐾))
11	5	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝐾 ∈ Poset)
12	6	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑆 ∈ dom 𝑈)
13	1	sselda 3936	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑆)
14	2, 3, 4, 11, 12, 13	luble 18419	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦(le‘𝐾)(𝑈‘𝑆))
15	14	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)(𝑈‘𝑆))
16		breq1 5111	. . . . . . 7 ⊢ (𝑦 = (𝑈‘𝑆) → (𝑦(le‘𝐾)𝑧 ↔ (𝑈‘𝑆)(le‘𝐾)𝑧))
17		simp3 1155	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧) → ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧)
18	9	3ad2ant1 1150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧) → (𝑈‘𝑆) ∈ 𝑇)
19	16, 17, 18	rspcdva 3581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧) → (𝑈‘𝑆)(le‘𝐾)𝑧)
20	19	3expia 1138	. . . . 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 5112	. . . . . . 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 5111	. . . . . . . 8 ⊢ (𝑥 = (𝑈‘𝑆) → (𝑥(le‘𝐾)𝑧 ↔ (𝑈‘𝑆)(le‘𝐾)𝑧))
25	24	imbi2d 343	. . . . . . 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 643	. . . . 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 3580	. . . 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 849	. . 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 264	. . . 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 49762	. . 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 725	. 2 ⊢ (𝜑 → 𝑇 ∈ dom 𝑈)
33	3, 2, 4, 5, 8, 10, 14, 19	poslubd 18473	. 2 ⊢ (𝜑 → (𝑈‘𝑇) = (𝑈‘𝑆))
34	32, 33	jca 520	1 ⊢ (𝜑 → (𝑇 ∈ dom 𝑈 ∧ (𝑈‘𝑇) = (𝑈‘𝑆)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ∃wrex 3088   ⊆ wss 3904   class class class wbr 5108  dom cdm 5660  ‘cfv 6536  Basecbs 17275  lecple 17323  Posetcpo 18369  lubclub 18371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-proset 18356  df-poset 18375  df-lub 18406

This theorem is used by:  lubprlem  49768