Theorem clsf2 38424

Description: The closure function is a map from the powerset of the base set to itself. This is less precise than clsf 20852. (Contributed by RP, 22-Apr-2021.)

Hypotheses

Ref	Expression
clselmap.x	|- X = U. J
clselmap.k	|- K = ( cls ` J )

Assertion

Ref	Expression
clsf2	|- ( J e. Top -> K : ~P X --> ~P X )

Proof of Theorem clsf2

Step	Hyp	Ref	Expression
1		clselmap.x	. . . 4 |- X = U. J
2	1	clsf 20852	. . 3 |- ( J e. Top -> ( cls ` J ) : ~P X --> ( Clsd ` J ) )
3		clselmap.k	. . . . 5 |- K = ( cls ` J )
4	3	feq1i 6036	. . . 4 |- ( K : ~P X --> ( Clsd ` J ) <-> ( cls ` J ) : ~P X --> ( Clsd ` J ) )
5		df-f 5892	. . . 4 |- ( K : ~P X --> ( Clsd ` J ) <-> ( K Fn ~P X /\ ran K C_ ( Clsd ` J ) ) )
6	4, 5	sylbb1 227	. . 3 |- ( ( cls ` J ) : ~P X --> ( Clsd ` J ) -> ( K Fn ~P X /\ ran K C_ ( Clsd ` J ) ) )
7	1	cldss2 20834	. . . . 5 |- ( Clsd ` J ) C_ ~P X
8		sstr2 3610	. . . . 5 |- ( ran K C_ ( Clsd ` J ) -> ( ( Clsd ` J ) C_ ~P X -> ran K C_ ~P X ) )
9	7, 8	mpi 20	. . . 4 |- ( ran K C_ ( Clsd ` J ) -> ran K C_ ~P X )
10	9	anim2i 593	. . 3 |- ( ( K Fn ~P X /\ ran K C_ ( Clsd ` J ) ) -> ( K Fn ~P X /\ ran K C_ ~P X ) )
11	2, 6, 10	3syl 18	. 2 |- ( J e. Top -> ( K Fn ~P X /\ ran K C_ ~P X ) )
12		df-f 5892	. 2 |- ( K : ~P X --> ~P X <-> ( K Fn ~P X /\ ran K C_ ~P X ) )
13	11, 12	sylibr 224	1 |- ( J e. Top -> K : ~P X --> ~P X )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888   Topctop 20698   Clsdccld 20820   clsccl 20822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-cld 20823  df-cls 20825

This theorem is referenced by:  clselmap  38425