Theorem lubeldm 17813

Description: Member of the domain of the least upper bound function of a poset. (Contributed by NM, 7-Sep-2018.)

Hypotheses

Ref	Expression
lubeldm.b	⊢ 𝐵 = (Base‘𝐾)
lubeldm.l	⊢ ≤ = (le‘𝐾)
lubeldm.u	⊢ 𝑈 = (lub‘𝐾)
lubeldm.p	⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
lubeldm.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)

Assertion

Ref	Expression
lubeldm	⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)))

Distinct variable groups:   𝑥,𝑧,𝐵   𝑥,𝑦,𝐾,𝑧   𝑥,𝑆,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝐵(𝑦)   𝑈(𝑥,𝑦,𝑧)   ≤ (𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem lubeldm

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lubeldm.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		lubeldm.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		lubeldm.u	. . . 4 ⊢ 𝑈 = (lub‘𝐾)
4		biid 264	. . . 4 ⊢ ((∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
5		lubeldm.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
6	1, 2, 3, 4, 5	lubdm 17811	. . 3 ⊢ (𝜑 → dom 𝑈 = {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))})
7	6	eleq2d 2816	. 2 ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))}))
8		raleq 3309	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥))
9		raleq 3309	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧))
10	9	imbi1d 345	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
11	10	ralbidv 3108	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
12	8, 11	anbi12d 634	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
13	12	reubidv 3291	. . . . 5 ⊢ (𝑠 = 𝑆 → (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
14		lubeldm.p	. . . . . 6 ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
15	14	reubii 3293	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
16	13, 15	bitr4di 292	. . . 4 ⊢ (𝑠 = 𝑆 → (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ∃!𝑥 ∈ 𝐵 𝜓))
17	16	elrab 3591	. . 3 ⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))} ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓))
18	1	fvexi 6709	. . . . 5 ⊢ 𝐵 ∈ V
19	18	elpw2 5223	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵)
20	19	anbi1i 627	. . 3 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓) ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓))
21	17, 20	bitri 278	. 2 ⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))} ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓))
22	7, 21	bitrdi 290	1 ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∀wral 3051  ∃!wreu 3053  {crab 3055   ⊆ wss 3853  𝒫 cpw 4499   class class class wbr 5039  dom cdm 5536  ‘cfv 6358  Basecbs 16666  lecple 16756  lubclub 17770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-lub 17806

This theorem is referenced by:  lubelss  17814  lubeu  17815  lubval  17816  lublecl  17821  lubeldm2  45866  joindm3  45879