Theorem fncld 23247

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 7741	. . . 4 ⊢ ∪ 𝑗 ∈ V
2	1	pwex 5345	. . 3 ⊢ 𝒫 ∪ 𝑗 ∈ V
3	2	rabex 5303	. 2 ⊢ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗} ∈ V
4		df-cld 23244	. 2 ⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗})
5	3, 4	fnmpti 6675	1 ⊢ Clsd Fn Top

This proof depends on syntax axioms:   ∈ wcel 2145  {crab 3412   ∖ cdif 3896  𝒫 cpw 4557  ∪ cuni 4867   Fn wfn 6528  Topctop 23118  Clsdccld 23241

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 2213  ax-ext 2732  ax-sep 5251  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-fun 6535  df-fn 6536  df-cld 23244

This theorem is used by:  cldrcl  23251  iscldtop  23320