Theorem flfval 20357

Description: Given a function from a filtered set to a topological space, define the set of limit points of the function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Assertion

Ref	Expression
flfval	|- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( ( J fLimf L ) ` F ) = ( J fLim ( ( X FilMap F ) ` L ) ) )

Proof of Theorem flfval

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		toponmax 19296	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. J )
2		filtop 20222	. . . . 5 |- ( L e. ( Fil ` Y ) -> Y e. L )
3		elmapg 7431	. . . . 5 |- ( ( X e. J /\ Y e. L ) -> ( F e. ( X ^m Y ) <-> F : Y --> X ) )
4	1, 2, 3	syl2an 477	. . . 4 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) -> ( F e. ( X ^m Y ) <-> F : Y --> X ) )
5	4	biimpar 485	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ F : Y --> X ) -> F e. ( X ^m Y ) )
6		flffval 20356	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) -> ( J fLimf L ) = ( f e. ( X ^m Y ) |-> ( J fLim ( ( X FilMap f ) ` L ) ) ) )
7	6	fveq1d 5854	. . . 4 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) -> ( ( J fLimf L ) ` F ) = ( ( f e. ( X ^m Y ) |-> ( J fLim ( ( X FilMap f ) ` L ) ) ) ` F ) )
8		oveq2 6285	. . . . . . 7 |- ( f = F -> ( X FilMap f ) = ( X FilMap F ) )
9	8	fveq1d 5854	. . . . . 6 |- ( f = F -> ( ( X FilMap f ) ` L ) = ( ( X FilMap F ) ` L ) )
10	9	oveq2d 6293	. . . . 5 |- ( f = F -> ( J fLim ( ( X FilMap f ) ` L ) ) = ( J fLim ( ( X FilMap F ) ` L ) ) )
11		eqid 2441	. . . . 5 |- ( f e. ( X ^m Y ) |-> ( J fLim ( ( X FilMap f ) ` L ) ) ) = ( f e. ( X ^m Y ) |-> ( J fLim ( ( X FilMap f ) ` L ) ) )
12		ovex 6305	. . . . 5 |- ( J fLim ( ( X FilMap F ) ` L ) ) e. _V
13	10, 11, 12	fvmpt 5937	. . . 4 |- ( F e. ( X ^m Y ) -> ( ( f e. ( X ^m Y ) |-> ( J fLim ( ( X FilMap f ) ` L ) ) ) ` F ) = ( J fLim ( ( X FilMap F ) ` L ) ) )
14	7, 13	sylan9eq 2502	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ F e. ( X ^m Y ) ) -> ( ( J fLimf L ) ` F ) = ( J fLim ( ( X FilMap F ) ` L ) ) )
15	5, 14	syldan 470	. 2 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ F : Y --> X ) -> ( ( J fLimf L ) ` F ) = ( J fLim ( ( X FilMap F ) ` L ) ) )
16	15	3impa 1190	1 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( ( J fLimf L ) ` F ) = ( J fLim ( ( X FilMap F ) ` L ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 972   = wceq 1381   e. wcel 1802   |-> cmpt 4491   -->wf 5570   ` cfv 5574  (class class class)co 6277   ^m cmap 7418  TopOnctopon 19262   Filcfil 20212   FilMap cfm 20300   fLim cflim 20301   fLimf cflf 20302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7420  df-fbas 18284  df-top 19266  df-topon 19269  df-fil 20213  df-flf 20307

This theorem is referenced by:  flfnei  20358  isflf  20360  hausflf  20364  flfcnp  20371  flfssfcf  20405  uffcfflf  20406  cnpfcf  20408  cnextcn  20433  tsmscls  20502  cnextucn  20672  cmetcaulem  21593  fmcncfil  27779