Theorem glbfun 18331

Description: The GLB is a function. (Contributed by NM, 9-Sep-2018.)

Hypothesis

Ref	Expression
glbfun.g	⊢ 𝐺 = (glb‘𝐾)

Assertion

Ref	Expression
glbfun	⊢ Fun 𝐺

Proof of Theorem glbfun

Dummy variables 𝑥 𝑠 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funmpt 6557	. . . 4 ⊢ Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))))
2		funres 6561	. . . 4 ⊢ (Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))) → Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))}))
3	1, 2	ax-mp 5	. . 3 ⊢ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))})
4		eqid 2730	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2730	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
6		glbfun.g	. . . . 5 ⊢ 𝐺 = (glb‘𝐾)
7		biid 261	. . . . 5 ⊢ ((∀𝑦 ∈ 𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)) ↔ (∀𝑦 ∈ 𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))
8		id 22	. . . . 5 ⊢ (𝐾 ∈ V → 𝐾 ∈ V)
9	4, 5, 6, 7, 8	glbfval 18329	. . . 4 ⊢ (𝐾 ∈ V → 𝐺 = ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))}))
10	9	funeqd 6541	. . 3 ⊢ (𝐾 ∈ V → (Fun 𝐺 ↔ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))})))
11	3, 10	mpbiri 258	. 2 ⊢ (𝐾 ∈ V → Fun 𝐺)
12		fun0 6584	. . 3 ⊢ Fun ∅
13		fvprc 6853	. . . . 5 ⊢ (¬ 𝐾 ∈ V → (glb‘𝐾) = ∅)
14	6, 13	eqtrid 2777	. . . 4 ⊢ (¬ 𝐾 ∈ V → 𝐺 = ∅)
15	14	funeqd 6541	. . 3 ⊢ (¬ 𝐾 ∈ V → (Fun 𝐺 ↔ Fun ∅))
16	12, 15	mpbiri 258	. 2 ⊢ (¬ 𝐾 ∈ V → Fun 𝐺)
17	11, 16	pm2.61i 182	1 ⊢ Fun 𝐺

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2708  ∀wral 3045  ∃!wreu 3354  Vcvv 3450  ∅c0 4299  𝒫 cpw 4566   class class class wbr 5110   ↦ cmpt 5191   ↾ cres 5643  Fun wfun 6508  ‘cfv 6514  ℩crio 7346  Basecbs 17186  lecple 17234  glbcglb 18278

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-glb 18313

This theorem is referenced by:  meetfval  18353  meetfval2  18354