Theorem fnmre 17511

Description: The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 22867 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 5312	. . . 4 ⊢ 𝒫 𝑥 ∈ V
2	1	pwex 5315	. . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V
3	2	rabex 5274	. 2 ⊢ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))} ∈ V
4		df-mre 17506	. 2 ⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
5	3, 4	fnmpti 6633	1 ⊢ Moore Fn V

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {crab 3390  Vcvv 3430  ∅c0 4274  𝒫 cpw 4542  ∩ cint 4890   Fn wfn 6485  Moorecmre 17502

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-sep 5231  ax-pow 5300  ax-pr 5368

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-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-fun 6492  df-fn 6493  df-mre 17506

This theorem is referenced by:  mreunirn  17521