Theorem fnmre 17681

Description: The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 23155 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 5346	. . . 4 ⊢ 𝒫 𝑥 ∈ V
2	1	pwex 5349	. . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V
3	2	rabex 5307	. 2 ⊢ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))} ∈ V
4		df-mre 17676	. 2 ⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
5	3, 4	fnmpti 6679	1 ⊢ Moore Fn V

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  {crab 3414  Vcvv 3453  ∅c0 4282  𝒫 cpw 4560  ∩ cint 4910   Fn wfn 6532  Moorecmre 17672

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pow 5334  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-fun 6539  df-fn 6540  df-mre 17676

This theorem is used by:  mreunirn  17691