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Theorem fncld 13601
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 4439 . . . 4 𝑗 ∈ V
21pwex 4184 . . 3 𝒫 𝑗 ∈ V
32rabex 4148 . 2 {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗} ∈ V
4 df-cld 13598 . 2 Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗})
53, 4fnmpti 5345 1 Clsd Fn Top
Colors of variables: wff set class
Syntax hints:  wcel 2148  {crab 2459  cdif 3127  𝒫 cpw 3576   cuni 3810   Fn wfn 5212  Topctop 13500  Clsdccld 13595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-fun 5219  df-fn 5220  df-cld 13598
This theorem is referenced by:  cldrcl  13605
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