Theorem fcfelbas 21840

Description: A cluster point of a function is in the base set of the topology. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
fcfelbas	|- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ A e. ( ( J fClusf L ) ` F ) ) -> A e. X )

Proof of Theorem fcfelbas

Step	Hyp	Ref	Expression
1		fcfval 21837	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( ( J fClusf L ) ` F ) = ( J fClus ( ( X FilMap F ) ` L ) ) )
2	1	eleq2d 2687	. . . 4 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( A e. ( ( J fClusf L ) ` F ) <-> A e. ( J fClus ( ( X FilMap F ) ` L ) ) ) )
3		eqid 2622	. . . . 5 |- U. J = U. J
4	3	fclselbas 21820	. . . 4 |- ( A e. ( J fClus ( ( X FilMap F ) ` L ) ) -> A e. U. J )
5	2, 4	syl6bi 243	. . 3 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( A e. ( ( J fClusf L ) ` F ) -> A e. U. J ) )
6	5	imp 445	. 2 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ A e. ( ( J fClusf L ) ` F ) ) -> A e. U. J )
7		simpl1 1064	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ A e. ( ( J fClusf L ) ` F ) ) -> J e. (TopOn ` X ) )
8		toponuni 20719	. . 3 |- ( J e. (TopOn ` X ) -> X = U. J )
9	7, 8	syl 17	. 2 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ A e. ( ( J fClusf L ) ` F ) ) -> X = U. J )
10	6, 9	eleqtrrd 2704	1 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ A e. ( ( J fClusf L ) ` F ) ) -> A e. X )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   U.cuni 4436   -->wf 5884   ` cfv 5888  (class class class)co 6650  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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-fbas 19743  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825  df-fil 21650  df-fcls 21745  df-fcf 21746

This theorem is referenced by: (None)