Theorem fnmre 17493

Description: The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 22809 for topologies. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Assertion

Ref	Expression
fnmre	⊢ Moore Fn V

Proof of Theorem fnmre

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

Step	Hyp	Ref	Expression
1		vpwex 5316	. . . 4 ⊢ 𝒫 𝑥 ∈ V
2	1	pwex 5319	. . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V
3	2	rabex 5278	. 2 ⊢ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))} ∈ V
4		df-mre 17488	. 2 ⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
5	3, 4	fnmpti 6625	1 ⊢ Moore Fn V

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3394  Vcvv 3436  ∅c0 4284  𝒫 cpw 4551  ∩ cint 4896   Fn wfn 6477  Moorecmre 17484

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 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-fun 6484  df-fn 6485  df-mre 17488

This theorem is referenced by:  mreunirn  17503