Theorem fncld 22525

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.

Step	Hyp	Ref	Expression
1		vuniex 7728	. . . 4 ⊢ ∪ 𝑗 ∈ V
2	1	pwex 5378	. . 3 ⊢ 𝒫 ∪ 𝑗 ∈ V
3	2	rabex 5332	. 2 ⊢ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗} ∈ V
4		df-cld 22522	. 2 ⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗})
5	3, 4	fnmpti 6693	1 ⊢ Clsd Fn Top

Syntax hints:   ∈ wcel 2106  {crab 3432   ∖ cdif 3945  𝒫 cpw 4602  ∪ cuni 4908   Fn wfn 6538  Topctop 22394  Clsdccld 22519

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-fun 6545  df-fn 6546  df-cld 22522

This theorem is referenced by:  cldrcl  22529  iscldtop  22598