Theorem fnmre 17094

Description: The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 21821 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 5270	. . . 4 ⊢ 𝒫 𝑥 ∈ V
2	1	pwex 5273	. . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V
3	2	rabex 5225	. 2 ⊢ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))} ∈ V
4		df-mre 17089	. 2 ⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
5	3, 4	fnmpti 6521	1 ⊢ Moore Fn V

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  {crab 3065  Vcvv 3408  ∅c0 4237  𝒫 cpw 4513  ∩ cint 4859   Fn wfn 6375  Moorecmre 17085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-fun 6382  df-fn 6383  df-mre 17089

This theorem is referenced by:  mreunirn  17104