Theorem fnmre 17559

Description: The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 22818 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 5335	. . . 4 ⊢ 𝒫 𝑥 ∈ V
2	1	pwex 5338	. . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V
3	2	rabex 5297	. 2 ⊢ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))} ∈ V
4		df-mre 17554	. 2 ⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
5	3, 4	fnmpti 6664	1 ⊢ Moore Fn V

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  {crab 3408  Vcvv 3450  ∅c0 4299  𝒫 cpw 4566  ∩ cint 4913   Fn wfn 6509  Moorecmre 17550

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-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-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  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-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-fun 6516  df-fn 6517  df-mre 17554

This theorem is referenced by:  mreunirn  17569