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Theorem hausflf 17692
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 17675 . . 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 17059 . . . . . 6  |-  ( J  e.  Haus  ->  J  e. 
Top )
4 hausflf.x . . . . . . 7  |-  X  = 
U. J
54toptopon 16671 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
63, 5sylib 188 . . . . 5  |-  ( J  e.  Haus  ->  J  e.  (TopOn `  X )
)
7 flfval 17685 . . . . 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 2350 . . 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 2178 . 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 1623    e. wcel 1684   E*wmo 2144   U.cuni 3827   -->wf 5251   ` cfv 5255  (class class class)co 5858   Topctop 16631  TopOnctopon 16632   Hauscha 17036   Filcfil 17540    FilMap cfm 17628    fLim cflim 17629    fLimf cflf 17630
This theorem is referenced by:  hausflf2  17693  haustsms  17818  limcmo  19232  nolimf  25619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-top 16636  df-topon 16639  df-nei 16835  df-haus 17043  df-fbas 17520  df-fil 17541  df-flim 17634  df-flf 17635
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