Theorem flfval 21794

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 20730	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. J )
2		filtop 21659	. . . . 5 |- ( L e. ( Fil ` Y ) -> Y e. L )
3		elmapg 7870	. . . . 5 |- ( ( X e. J /\ Y e. L ) -> ( F e. ( X ^m Y ) <-> F : Y --> X ) )
4	1, 2, 3	syl2an 494	. . . 4 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) -> ( F e. ( X ^m Y ) <-> F : Y --> X ) )
5	4	biimpar 502	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) ) /\ F : Y --> X ) -> F e. ( X ^m Y ) )
6		flffval 21793	. . . . 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 6193	. . . 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 6658	. . . . . . 7 |- ( f = F -> ( X FilMap f ) = ( X FilMap F ) )
9	8	fveq1d 6193	. . . . . 6 |- ( f = F -> ( ( X FilMap f ) ` L ) = ( ( X FilMap F ) ` L ) )
10	9	oveq2d 6666	. . . . 5 |- ( f = F -> ( J fLim ( ( X FilMap f ) ` L ) ) = ( J fLim ( ( X FilMap F ) ` L ) ) )
11		eqid 2622	. . . . 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 6678	. . . . 5 |- ( J fLim ( ( X FilMap F ) ` L ) ) e. _V
13	10, 11, 12	fvmpt 6282	. . . 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 2676	. . 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 487	. 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 1259	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857  TopOnctopon 20715   Filcfil 21649   FilMap cfm 21737   fLim cflim 21738   fLimf cflf 21739

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-flf 21744

This theorem is referenced by:  flfnei  21795  isflf  21797  hausflf  21801  flfcnp  21808  flfssfcf  21842  uffcfflf  21843  cnpfcf  21845  cnextcn  21871  tsmscls  21941  cnextucn  22107  cmetcaulem  23086  fmcncfil  29977