Theorem glbfun 18373

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 6573	. . . 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 6577	. . . 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 2735	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2735	. . . . 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 18371	. . . 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 6557	. . 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 6600	. . 3 ⊢ Fun ∅
13		fvprc 6867	. . . . 5 ⊢ (¬ 𝐾 ∈ V → (glb‘𝐾) = ∅)
14	6, 13	eqtrid 2782	. . . 4 ⊢ (¬ 𝐾 ∈ V → 𝐺 = ∅)
15	14	funeqd 6557	. . 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 2108  {cab 2713  ∀wral 3051  ∃!wreu 3357  Vcvv 3459  ∅c0 4308  𝒫 cpw 4575   class class class wbr 5119   ↦ cmpt 5201   ↾ cres 5656  Fun wfun 6524  ‘cfv 6530  ℩crio 7359  Basecbs 17226  lecple 17276  glbcglb 18320

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-glb 18355

This theorem is referenced by:  meetfval  18395  meetfval2  18396