Theorem flfval 19705

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 18675	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. J )
2		filtop 19570	. . . . 5 |- ( L e. ( Fil ` Y ) -> Y e. L )
3		elmapg 7340	. . . . 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 19704	. . . . 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 5804	. . . 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 6211	. . . . . . 7 |- ( f = F -> ( X FilMap f ) = ( X FilMap F ) )
9	8	fveq1d 5804	. . . . . 6 |- ( f = F -> ( ( X FilMap f ) ` L ) = ( ( X FilMap F ) ` L ) )
10	9	oveq2d 6219	. . . . 5 |- ( f = F -> ( J fLim ( ( X FilMap f ) ` L ) ) = ( J fLim ( ( X FilMap F ) ` L ) ) )
11		eqid 2454	. . . . 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 6228	. . . . 5 |- ( J fLim ( ( X FilMap F ) ` L ) ) e. _V
13	10, 11, 12	fvmpt 5886	. . . 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 2515	. . 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 1183	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 965   = wceq 1370   e. wcel 1758   |-> cmpt 4461   -->wf 5525   ` cfv 5529  (class class class)co 6203   ^m cmap 7327  TopOnctopon 18641   Filcfil 19560   FilMap cfm 19648   fLim cflim 19649   fLimf cflf 19650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-map 7329  df-fbas 17949  df-top 18645  df-topon 18648  df-fil 19561  df-flf 19655

This theorem is referenced by:  flfnei  19706  isflf  19708  hausflf  19712  flfcnp  19719  flfssfcf  19753  uffcfflf  19754  cnpfcf  19756  cnextcn  19781  tsmscls  19850  cnextucn  20020  cmetcaulem  20941  fmcncfil  26529