Theorem lubsscl 48948

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 2729	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2729	. . . . 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 18313	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐾))
8	1, 7	sstrd 3957	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐾))
9		lubsscl.x	. . . . 5 ⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝑇)
10	8, 9	sseldd 3947	. . . 4 ⊢ (𝜑 → (𝑈‘𝑆) ∈ (Base‘𝐾))
11	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝐾 ∈ Poset)
12	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑆 ∈ dom 𝑈)
13	1	sselda 3946	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑆)
14	2, 3, 4, 11, 12, 13	luble 18318	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦(le‘𝐾)(𝑈‘𝑆))
15	14	ralrimiva 3125	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)(𝑈‘𝑆))
16		breq1 5110	. . . . . . 7 ⊢ (𝑦 = (𝑈‘𝑆) → (𝑦(le‘𝐾)𝑧 ↔ (𝑈‘𝑆)(le‘𝐾)𝑧))
17		simp3 1138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧) → ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧)
18	9	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧) → (𝑈‘𝑆) ∈ 𝑇)
19	16, 17, 18	rspcdva 3589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧) → (𝑈‘𝑆)(le‘𝐾)𝑧)
20	19	3expia 1121	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾)) → (∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → (𝑈‘𝑆)(le‘𝐾)𝑧))
21	20	ralrimiva 3125	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → (𝑈‘𝑆)(le‘𝐾)𝑧))
22		breq2 5111	. . . . . . 7 ⊢ (𝑥 = (𝑈‘𝑆) → (𝑦(le‘𝐾)𝑥 ↔ 𝑦(le‘𝐾)(𝑈‘𝑆)))
23	22	ralbidv 3156	. . . . . 6 ⊢ (𝑥 = (𝑈‘𝑆) → (∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑥 ↔ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)(𝑈‘𝑆)))
24		breq1 5110	. . . . . . . 8 ⊢ (𝑥 = (𝑈‘𝑆) → (𝑥(le‘𝐾)𝑧 ↔ (𝑈‘𝑆)(le‘𝐾)𝑧))
25	24	imbi2d 340	. . . . . . 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 3156	. . . . . 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 632	. . . . 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 3588	. . . 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 836	. . 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 261	. . . 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 48944	. . 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 713	. 2 ⊢ (𝜑 → 𝑇 ∈ dom 𝑈)
33	3, 2, 4, 5, 8, 10, 14, 19	poslubd 18372	. 2 ⊢ (𝜑 → (𝑈‘𝑇) = (𝑈‘𝑆))
34	32, 33	jca 511	1 ⊢ (𝜑 → (𝑇 ∈ dom 𝑈 ∧ (𝑈‘𝑇) = (𝑈‘𝑆)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914   class class class wbr 5107  dom cdm 5638  ‘cfv 6511  Basecbs 17179  lecple 17227  Posetcpo 18268  lubclub 18270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-proset 18255  df-poset 18274  df-lub 18305

This theorem is referenced by:  lubprlem  48950