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Theorem hausflf 17708
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 17691 . . 3  |-  ( J  e.  Haus  ->  E* x  x  e.  ( J  fLim  ( ( X  FilMap  F ) `  L ) ) )
213ad2ant1 976 . 2  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  E* x  x  e.  ( J  fLim  ( ( X 
FilMap  F ) `  L
) ) )
3 haustop 17075 . . . . . 6  |-  ( J  e.  Haus  ->  J  e. 
Top )
4 hausflf.x . . . . . . 7  |-  X  = 
U. J
54toptopon 16687 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
63, 5sylib 188 . . . . 5  |-  ( J  e.  Haus  ->  J  e.  (TopOn `  X )
)
7 flfval 17701 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
86, 7syl3an1 1215 . . . 4  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
98eleq2d 2363 . . 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 2191 . 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 223 1  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  E* x  x  e.  (
( J  fLimf  L ) `
 F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   E*wmo 2157   U.cuni 3843   -->wf 5267   ` cfv 5271  (class class class)co 5874   Topctop 16647  TopOnctopon 16648   Hauscha 17052   Filcfil 17556    FilMap cfm 17644    fLim cflim 17645    fLimf cflf 17646
This theorem is referenced by:  hausflf2  17709  haustsms  17834  limcmo  19248  nolimf  25722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-top 16652  df-topon 16655  df-nei 16851  df-haus 17059  df-fbas 17536  df-fil 17557  df-flim 17650  df-flf 17651
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