Theorem fncld 14821

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 4535	. . . 4 |- U. j e. _V
2	1	pwex 4273	. . 3 |- ~P U. j e. _V
3	2	rabex 4234	. 2 |- { x e. ~P U. j | ( U. j \ x ) e. j } e. _V
4		df-cld 14818	. 2 |- Clsd = ( j e. Top |-> { x e. ~P U. j | ( U. j \ x ) e. j } )
5	3, 4	fnmpti 5461	1 |- Clsd Fn Top

Syntax hints:   e. wcel 2202   {crab 2514   \ cdif 3197   ~Pcpw 3652   U.cuni 3893   Fn wfn 5321   Topctop 14720   Clsdccld 14815

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-fun 5328  df-fn 5329  df-cld 14818

This theorem is referenced by:  cldrcl  14825