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Theorem hausflf2 18031
Description: If a convergent function has its values in a Hausdorff space, then it has a unique limit. (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
hausflf2  |-  ( ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( ( J  fLimf  L ) `  F )  =/=  (/) )  ->  (
( J  fLimf  L ) `
 F )  ~~  1o )

Proof of Theorem hausflf2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 n0 3638 . . 3  |-  ( ( ( J  fLimf  L ) `
 F )  =/=  (/) 
<->  E. x  x  e.  ( ( J  fLimf  L ) `  F ) )
21biimpi 188 . 2  |-  ( ( ( J  fLimf  L ) `
 F )  =/=  (/)  ->  E. x  x  e.  ( ( J  fLimf  L ) `  F ) )
3 hausflf.x . . 3  |-  X  = 
U. J
43hausflf 18030 . 2  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  E* x  x  e.  (
( J  fLimf  L ) `
 F ) )
5 euen1b 7179 . . . 4  |-  ( ( ( J  fLimf  L ) `
 F )  ~~  1o 
<->  E! x  x  e.  ( ( J  fLimf  L ) `  F ) )
6 eu5 2320 . . . 4  |-  ( E! x  x  e.  ( ( J  fLimf  L ) `
 F )  <->  ( E. x  x  e.  (
( J  fLimf  L ) `
 F )  /\  E* x  x  e.  ( ( J  fLimf  L ) `  F ) ) )
75, 6bitr2i 243 . . 3  |-  ( ( E. x  x  e.  ( ( J  fLimf  L ) `  F )  /\  E* x  x  e.  ( ( J 
fLimf  L ) `  F
) )  <->  ( ( J  fLimf  L ) `  F )  ~~  1o )
87biimpi 188 . 2  |-  ( ( E. x  x  e.  ( ( J  fLimf  L ) `  F )  /\  E* x  x  e.  ( ( J 
fLimf  L ) `  F
) )  ->  (
( J  fLimf  L ) `
 F )  ~~  1o )
92, 4, 8syl2anr 466 1  |-  ( ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( ( J  fLimf  L ) `  F )  =/=  (/) )  ->  (
( J  fLimf  L ) `
 F )  ~~  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   E!weu 2282   E*wmo 2283    =/= wne 2600   (/)c0 3629   U.cuni 4016   class class class wbr 4213   -->wf 5451   ` cfv 5455  (class class class)co 6082   1oc1o 6718    ~~ cen 7107   Hauscha 17373   Filcfil 17878    fLimf cflf 17968
This theorem is referenced by:  cnextfvval  18097  cnextcn  18099  cnextfres  18100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-suc 4588  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1o 6725  df-map 7021  df-en 7111  df-fbas 16700  df-top 16964  df-topon 16967  df-nei 17163  df-haus 17380  df-fil 17879  df-flim 17972  df-flf 17973
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