Theorem lubeldm 18272

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 261	. . . 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 18270	. . 3 ⊢ (𝜑 → dom 𝑈 = {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))})
7	6	eleq2d 2820	. 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 3291	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥))
9		raleq 3291	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧))
10	9	imbi1d 341	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
11	10	ralbidv 3157	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
12	8, 11	anbi12d 632	. . . . . 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 3364	. . . . 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 3357	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
16	13, 15	bitr4di 289	. . . 4 ⊢ (𝑠 = 𝑆 → (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ∃!𝑥 ∈ 𝐵 𝜓))
17	16	elrab 3644	. . 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 6846	. . . . 5 ⊢ 𝐵 ∈ V
19	18	elpw2 5277	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵)
20	19	anbi1i 624	. . 3 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓) ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓))
21	17, 20	bitri 275	. 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 287	1 ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃!wreu 3346  {crab 3397   ⊆ wss 3899  𝒫 cpw 4552   class class class wbr 5096  dom cdm 5622  ‘cfv 6490  Basecbs 17134  lecple 17182  lubclub 18230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-lub 18265

This theorem is referenced by:  lubelss  18273  lubeu  18274  lubval  18275  lublecl  18280  lubeldm2  49143  joindm3  49156