Theorem glbeldm 18318

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

Hypotheses

Ref	Expression
glbeldm.b	⊢ 𝐵 = (Base‘𝐾)
glbeldm.l	⊢ ≤ = (le‘𝐾)
glbeldm.g	⊢ 𝐺 = (glb‘𝐾)
glbeldm.p	⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
glbeldm.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)

Assertion

Ref	Expression
glbeldm	⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (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)   𝐺(𝑥,𝑦,𝑧)   ≤ (𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem glbeldm

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		glbeldm.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		glbeldm.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		glbeldm.g	. . . 4 ⊢ 𝐺 = (glb‘𝐾)
4		biid 260	. . . 4 ⊢ ((∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
5		glbeldm.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
6	1, 2, 3, 4, 5	glbdm 18316	. . 3 ⊢ (𝜑 → dom 𝐺 = {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))})
7	6	eleq2d 2819	. 2 ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))}))
8		raleq 3322	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦))
9		raleq 3322	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦))
10	9	imbi1d 341	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
11	10	ralbidv 3177	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
12	8, 11	anbi12d 631	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
13	12	reubidv 3394	. . . . 5 ⊢ (𝑠 = 𝑆 → (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
14		glbeldm.p	. . . . . 6 ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
15	14	reubii 3385	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
16	13, 15	bitr4di 288	. . . 4 ⊢ (𝑠 = 𝑆 → (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ∃!𝑥 ∈ 𝐵 𝜓))
17	16	elrab 3683	. . 3 ⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))} ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓))
18	1	fvexi 6905	. . . . 5 ⊢ 𝐵 ∈ V
19	18	elpw2 5345	. . . 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 274	. 2 ⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))} ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓))
22	7, 21	bitrdi 286	1 ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃!wreu 3374  {crab 3432   ⊆ wss 3948  𝒫 cpw 4602   class class class wbr 5148  dom cdm 5676  ‘cfv 6543  Basecbs 17143  lecple 17203  glbcglb 18262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-glb 18299

This theorem is referenced by:  glbelss  18319  glbeu  18320  glbval  18321  glbeldm2  47580  meetdm3  47594