Theorem fcfval 21837

Description: The set of cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
fcfval	|- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( ( J fClusf L ) ` F ) = ( J fClus ( ( X FilMap F ) ` L ) ) )

Proof of Theorem fcfval

Dummy variables f g j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fcf 21746	. . . . 5 |- fClusf = ( j e. Top , f e. U. ran Fil |-> ( g e. ( U. j ^m U. f ) |-> ( j fClus ( ( U. j FilMap g ) ` f ) ) ) )
2	1	a1i 11	. . . 4 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) -> fClusf = ( j e. Top , f e. U. ran Fil |-> ( g e. ( U. j ^m U. f ) |-> ( j fClus ( ( U. j FilMap g ) ` f ) ) ) ) )
3		simprl 794	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ ( j = J /\ f = L ) ) -> j = J )
4	3	unieqd 4446	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ ( j = J /\ f = L ) ) -> U. j = U. J )
5		toponuni 20719	. . . . . . . 8 |- ( J e. (TopOn ` X ) -> X = U. J )
6	5	ad2antrr 762	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ ( j = J /\ f = L ) ) -> X = U. J )
7	4, 6	eqtr4d 2659	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ ( j = J /\ f = L ) ) -> U. j = X )
8		unieq 4444	. . . . . . . 8 |- ( f = L -> U. f = U. L )
9	8	ad2antll 765	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ ( j = J /\ f = L ) ) -> U. f = U. L )
10		filunibas 21685	. . . . . . . 8 |- ( L e. ( Fil ` Y ) -> U. L = Y )
11	10	ad2antlr 763	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ ( j = J /\ f = L ) ) -> U. L = Y )
12	9, 11	eqtrd 2656	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ ( j = J /\ f = L ) ) -> U. f = Y )
13	7, 12	oveq12d 6668	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ ( j = J /\ f = L ) ) -> ( U. j ^m U. f ) = ( X ^m Y ) )
14	7	oveq1d 6665	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ ( j = J /\ f = L ) ) -> ( U. j FilMap g ) = ( X FilMap g ) )
15		simprr 796	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ ( j = J /\ f = L ) ) -> f = L )
16	14, 15	fveq12d 6197	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ ( j = J /\ f = L ) ) -> ( ( U. j FilMap g ) ` f ) = ( ( X FilMap g ) ` L ) )
17	3, 16	oveq12d 6668	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ ( j = J /\ f = L ) ) -> ( j fClus ( ( U. j FilMap g ) ` f ) ) = ( J fClus ( ( X FilMap g ) ` L ) ) )
18	13, 17	mpteq12dv 4733	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ ( j = J /\ f = L ) ) -> ( g e. ( U. j ^m U. f ) |-> ( j fClus ( ( U. j FilMap g ) ` f ) ) ) = ( g e. ( X ^m Y ) |-> ( J fClus ( ( X FilMap g ) ` L ) ) ) )
19		topontop 20718	. . . . 5 |- ( J e. (TopOn ` X ) -> J e. Top )
20	19	adantr 481	. . . 4 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) -> J e. Top )
21		fvssunirn 6217	. . . . . 6 |- ( Fil ` Y ) C_ U. ran Fil
22	21	sseli 3599	. . . . 5 |- ( L e. ( Fil ` Y ) -> L e. U. ran Fil )
23	22	adantl 482	. . . 4 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) -> L e. U. ran Fil )
24		ovex 6678	. . . . . 6 |- ( X ^m Y ) e. _V
25	24	mptex 6486	. . . . 5 |- ( g e. ( X ^m Y ) |-> ( J fClus ( ( X FilMap g ) ` L ) ) ) e. _V
26	25	a1i 11	. . . 4 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) -> ( g e. ( X ^m Y ) |-> ( J fClus ( ( X FilMap g ) ` L ) ) ) e. _V )
27	2, 18, 20, 23, 26	ovmpt2d 6788	. . 3 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) -> ( J fClusf L ) = ( g e. ( X ^m Y ) |-> ( J fClus ( ( X FilMap g ) ` L ) ) ) )
28	27	3adant3 1081	. 2 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( J fClusf L ) = ( g e. ( X ^m Y ) |-> ( J fClus ( ( X FilMap g ) ` L ) ) ) )
29		simpr 477	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ g = F ) -> g = F )
30	29	oveq2d 6666	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ g = F ) -> ( X FilMap g ) = ( X FilMap F ) )
31	30	fveq1d 6193	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ g = F ) -> ( ( X FilMap g ) ` L ) = ( ( X FilMap F ) ` L ) )
32	31	oveq2d 6666	. 2 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ g = F ) -> ( J fClus ( ( X FilMap g ) ` L ) ) = ( J fClus ( ( X FilMap F ) ` L ) ) )
33		toponmax 20730	. . . 4 |- ( J e. (TopOn ` X ) -> X e. J )
34		filtop 21659	. . . 4 |- ( L e. ( Fil ` Y ) -> Y e. L )
35		elmapg 7870	. . . 4 |- ( ( X e. J /\ Y e. L ) -> ( F e. ( X ^m Y ) <-> F : Y --> X ) )
36	33, 34, 35	syl2an 494	. . 3 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) -> ( F e. ( X ^m Y ) <-> F : Y --> X ) )
37	36	biimp3ar 1433	. 2 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> F e. ( X ^m Y ) )
38		ovexd 6680	. 2 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( J fClus ( ( X FilMap F ) ` L ) ) e. _V )
39	28, 32, 37, 38	fvmptd 6288	1 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( ( J fClusf L ) ` F ) = ( J fClus ( ( X FilMap F ) ` L ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   U.cuni 4436   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   Topctop 20698  TopOnctopon 20715   Filcfil 21649   FilMap cfm 21737   fClus cfcls 21740   fClusf cfcf 21741

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-nel 2898  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-iun 4522  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-fbas 19743  df-top 20699  df-topon 20716  df-fil 21650  df-fcf 21746

This theorem is referenced by:  isfcf  21838  fcfelbas  21840  flfssfcf  21842  uffcfflf  21843  cnpfcfi  21844  cnpfcf  21845