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Theorem hausflf 17990
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 17973 . . 3  |-  ( J  e.  Haus  ->  E* x  x  e.  ( J  fLim  ( ( X  FilMap  F ) `  L ) ) )
213ad2ant1 978 . 2  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  E* x  x  e.  ( J  fLim  ( ( X 
FilMap  F ) `  L
) ) )
3 haustop 17357 . . . . . 6  |-  ( J  e.  Haus  ->  J  e. 
Top )
4 hausflf.x . . . . . . 7  |-  X  = 
U. J
54toptopon 16961 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
63, 5sylib 189 . . . . 5  |-  ( J  e.  Haus  ->  J  e.  (TopOn `  X )
)
7 flfval 17983 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
86, 7syl3an1 1217 . . . 4  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
98eleq2d 2479 . . 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 2297 . 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 224 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 936    = wceq 1649    e. wcel 1721   E*wmo 2263   U.cuni 3983   -->wf 5417   ` cfv 5421  (class class class)co 6048   Topctop 16921  TopOnctopon 16922   Hauscha 17334   Filcfil 17838    FilMap cfm 17926    fLim cflim 17927    fLimf cflf 17928
This theorem is referenced by:  hausflf2  17991  cnextfun  18056  haustsms  18126  limcmo  19730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-fbas 16662  df-top 16926  df-topon 16929  df-nei 17125  df-haus 17341  df-fil 17839  df-flim 17932  df-flf 17933
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