Theorem lubsscl 48640

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 2740	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2740	. . . . 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 18424	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐾))
8	1, 7	sstrd 4019	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐾))
9		lubsscl.x	. . . . 5 ⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝑇)
10	8, 9	sseldd 4009	. . . 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 4008	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑆)
14	2, 3, 4, 11, 12, 13	luble 18429	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦(le‘𝐾)(𝑈‘𝑆))
15	14	ralrimiva 3152	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)(𝑈‘𝑆))
16		breq1 5169	. . . . . . 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 3636	. . . . . 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 3152	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑧 → (𝑈‘𝑆)(le‘𝐾)𝑧))
22		breq2 5170	. . . . . . 7 ⊢ (𝑥 = (𝑈‘𝑆) → (𝑦(le‘𝐾)𝑥 ↔ 𝑦(le‘𝐾)(𝑈‘𝑆)))
23	22	ralbidv 3184	. . . . . 6 ⊢ (𝑥 = (𝑈‘𝑆) → (∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)𝑥 ↔ ∀𝑦 ∈ 𝑇 𝑦(le‘𝐾)(𝑈‘𝑆)))
24		breq1 5169	. . . . . . . 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 3184	. . . . . 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 3635	. . . 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 48636	. . 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 712	. 2 ⊢ (𝜑 → 𝑇 ∈ dom 𝑈)
33	3, 2, 4, 5, 8, 10, 14, 19	poslubd 18483	. 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 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976   class class class wbr 5166  dom cdm 5700  ‘cfv 6573  Basecbs 17258  lecple 17318  Posetcpo 18377  lubclub 18379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-proset 18365  df-poset 18383  df-lub 18416

This theorem is referenced by:  lubprlem  48642