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Theorem fncld 21641
 Description: The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fncld Clsd Fn Top

Proof of Theorem fncld
Dummy variables 𝑥 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vuniex 7452 . . . 4 𝑗 ∈ V
21pwex 5247 . . 3 𝒫 𝑗 ∈ V
32rabex 5200 . 2 {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗} ∈ V
4 df-cld 21638 . 2 Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗})
53, 4fnmpti 6466 1 Clsd Fn Top
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2111  {crab 3110   ∖ cdif 3878  𝒫 cpw 4497  ∪ cuni 4801   Fn wfn 6322  Topctop 21512  Clsdccld 21635 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-fun 6329  df-fn 6330  df-cld 21638 This theorem is referenced by:  cldrcl  21645  iscldtop  21714
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