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Theorem fncld 12445
 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 4393 . . . 4 𝑗 ∈ V
21pwex 4139 . . 3 𝒫 𝑗 ∈ V
32rabex 4104 . 2 {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗} ∈ V
4 df-cld 12442 . 2 Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗})
53, 4fnmpti 5291 1 Clsd Fn Top
 Colors of variables: wff set class Syntax hints:   ∈ wcel 2125  {crab 2436   ∖ cdif 3095  𝒫 cpw 3539  ∪ cuni 3768   Fn wfn 5158  Topctop 12342  Clsdccld 12439 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-rab 2441  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-fun 5165  df-fn 5166  df-cld 12442 This theorem is referenced by:  cldrcl  12449
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