Theorem glbfun 18286

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 6530	. . . 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 6534	. . . 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 2736	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2736	. . . . 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 18284	. . . 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 6514	. . 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 6557	. . 3 ⊢ Fun ∅
13		fvprc 6826	. . . . 5 ⊢ (¬ 𝐾 ∈ V → (glb‘𝐾) = ∅)
14	6, 13	eqtrid 2783	. . . 4 ⊢ (¬ 𝐾 ∈ V → 𝐺 = ∅)
15	14	funeqd 6514	. . 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 1541   ∈ wcel 2113  {cab 2714  ∀wral 3051  ∃!wreu 3348  Vcvv 3440  ∅c0 4285  𝒫 cpw 4554   class class class wbr 5098   ↦ cmpt 5179   ↾ cres 5626  Fun wfun 6486  ‘cfv 6492  ℩crio 7314  Basecbs 17136  lecple 17184  glbcglb 18233

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-glb 18268

This theorem is referenced by:  meetfval  18308  meetfval2  18309