Theorem uffcfflf 21843

Description: If the domain filter is an ultrafilter, the cluster points of the function are the limit points. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
uffcfflf	|- ( ( J e. (TopOn ` X ) /\ L e. ( UFil ` Y ) /\ F : Y --> X ) -> ( ( J fClusf L ) ` F ) = ( ( J fLimf L ) ` F ) )

Proof of Theorem uffcfflf

Step	Hyp	Ref	Expression
1		toponmax 20730	. . . 4 |- ( J e. (TopOn ` X ) -> X e. J )
2		fmufil 21763	. . . 4 |- ( ( X e. J /\ L e. ( UFil ` Y ) /\ F : Y --> X ) -> ( ( X FilMap F ) ` L ) e. ( UFil ` X ) )
3	1, 2	syl3an1 1359	. . 3 |- ( ( J e. (TopOn ` X ) /\ L e. ( UFil ` Y ) /\ F : Y --> X ) -> ( ( X FilMap F ) ` L ) e. ( UFil ` X ) )
4		uffclsflim 21835	. . 3 |- ( ( ( X FilMap F ) ` L ) e. ( UFil ` X ) -> ( J fClus ( ( X FilMap F ) ` L ) ) = ( J fLim ( ( X FilMap F ) ` L ) ) )
5	3, 4	syl 17	. 2 |- ( ( J e. (TopOn ` X ) /\ L e. ( UFil ` Y ) /\ F : Y --> X ) -> ( J fClus ( ( X FilMap F ) ` L ) ) = ( J fLim ( ( X FilMap F ) ` L ) ) )
6		ufilfil 21708	. . 3 |- ( L e. ( UFil ` Y ) -> L e. ( Fil ` Y ) )
7		fcfval 21837	. . 3 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( ( J fClusf L ) ` F ) = ( J fClus ( ( X FilMap F ) ` L ) ) )
8	6, 7	syl3an2 1360	. 2 |- ( ( J e. (TopOn ` X ) /\ L e. ( UFil ` Y ) /\ F : Y --> X ) -> ( ( J fClusf L ) ` F ) = ( J fClus ( ( X FilMap F ) ` L ) ) )
9		flfval 21794	. . 3 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( ( J fLimf L ) ` F ) = ( J fLim ( ( X FilMap F ) ` L ) ) )
10	6, 9	syl3an2 1360	. 2 |- ( ( J e. (TopOn ` X ) /\ L e. ( UFil ` Y ) /\ F : Y --> X ) -> ( ( J fLimf L ) ` F ) = ( J fLim ( ( X FilMap F ) ` L ) ) )
11	5, 8, 10	3eqtr4d 2666	1 |- ( ( J e. (TopOn ` X ) /\ L e. ( UFil ` Y ) /\ F : Y --> X ) -> ( ( J fClusf L ) ` F ) = ( ( J fLimf L ) ` F ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   -->wf 5884   ` cfv 5888  (class class class)co 6650  TopOnctopon 20715   Filcfil 21649   UFilcufil 21703   FilMap cfm 21737   fLim cflim 21738   fLimf cflf 21739   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-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  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-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-fi 8317  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-fil 21650  df-ufil 21705  df-fm 21742  df-flim 21743  df-flf 21744  df-fcls 21745  df-fcf 21746

This theorem is referenced by: (None)