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Theorem hausflf 20943
Description: If a function has its values in a Hausdorff space, then it has at most one limit value. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
hausflf.x  |-  X  = 
U. J
Assertion
Ref Expression
hausflf  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  E* x  x  e.  (
( J  fLimf  L ) `
 F ) )
Distinct variable groups:    x, F    x, J    x, L    x, X    x, Y

Proof of Theorem hausflf
StepHypRef Expression
1 hausflimi 20926 . . 3  |-  ( J  e.  Haus  ->  E* x  x  e.  ( J  fLim  ( ( X  FilMap  F ) `  L ) ) )
213ad2ant1 1026 . 2  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  E* x  x  e.  ( J  fLim  ( ( X 
FilMap  F ) `  L
) ) )
3 haustop 20278 . . . . . 6  |-  ( J  e.  Haus  ->  J  e. 
Top )
4 hausflf.x . . . . . . 7  |-  X  = 
U. J
54toptopon 19879 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
63, 5sylib 199 . . . . 5  |-  ( J  e.  Haus  ->  J  e.  (TopOn `  X )
)
7 flfval 20936 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
86, 7syl3an1 1297 . . . 4  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
98eleq2d 2499 . . 3  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
x  e.  ( ( J  fLimf  L ) `  F )  <->  x  e.  ( J  fLim  ( ( X  FilMap  F ) `  L ) ) ) )
109mobidv 2289 . 2  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  ( E* x  x  e.  ( ( J  fLimf  L ) `  F )  <->  E* x  x  e.  ( J  fLim  ( ( X  FilMap  F ) `  L ) ) ) )
112, 10mpbird 235 1  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  E* x  x  e.  (
( J  fLimf  L ) `
 F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1870   E*wmo 2267   U.cuni 4222   -->wf 5597   ` cfv 5601  (class class class)co 6305   Topctop 19848  TopOnctopon 19849   Hauscha 20255   Filcfil 20791    FilMap cfm 20879    fLim cflim 20880    fLimf cflf 20881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482  df-fbas 18902  df-top 19852  df-topon 19854  df-nei 20045  df-haus 20262  df-fil 20792  df-flim 20885  df-flf 20886
This theorem is referenced by:  hausflf2  20944  cnextfun  21010  haustsms  21081  limcmo  22714
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