Theorem glbfun 18298

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 6538	. . . 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 6542	. . . 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 2737	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2737	. . . . 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 18296	. . . 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 6522	. . 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 6565	. . 3 ⊢ Fun ∅
13		fvprc 6834	. . . . 5 ⊢ (¬ 𝐾 ∈ V → (glb‘𝐾) = ∅)
14	6, 13	eqtrid 2784	. . . 4 ⊢ (¬ 𝐾 ∈ V → 𝐺 = ∅)
15	14	funeqd 6522	. . 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 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃!wreu 3350  Vcvv 3442  ∅c0 4287  𝒫 cpw 4556   class class class wbr 5100   ↦ cmpt 5181   ↾ cres 5634  Fun wfun 6494  ‘cfv 6500  ℩crio 7324  Basecbs 17148  lecple 17196  glbcglb 18245

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-glb 18280

This theorem is referenced by:  meetfval  18320  meetfval2  18321