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Theorem hausflf2 20935
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 3777 . . 3  |-  ( ( ( J  fLimf  L ) `
 F )  =/=  (/) 
<->  E. x  x  e.  ( ( J  fLimf  L ) `  F ) )
21biimpi 197 . 2  |-  ( ( ( J  fLimf  L ) `
 F )  =/=  (/)  ->  E. x  x  e.  ( ( J  fLimf  L ) `  F ) )
3 hausflf.x . . 3  |-  X  = 
U. J
43hausflf 20934 . 2  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  E* x  x  e.  (
( J  fLimf  L ) `
 F ) )
5 euen1b 7647 . . . 4  |-  ( ( ( J  fLimf  L ) `
 F )  ~~  1o 
<->  E! x  x  e.  ( ( J  fLimf  L ) `  F ) )
6 eu5 2294 . . . 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 253 . . 3  |-  ( ( E. x  x  e.  ( ( J  fLimf  L ) `  F )  /\  E* x  x  e.  ( ( J 
fLimf  L ) `  F
) )  <->  ( ( J  fLimf  L ) `  F )  ~~  1o )
87biimpi 197 . 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 480 1  |-  ( ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( ( J  fLimf  L ) `  F )  =/=  (/) )  ->  (
( J  fLimf  L ) `
 F )  ~~  1o )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1870   E!weu 2266   E*wmo 2267    =/= wne 2625   (/)c0 3767   U.cuni 4222   class class class wbr 4426   -->wf 5597   ` cfv 5601  (class class class)co 6305   1oc1o 7183    ~~ cen 7574   Hauscha 20246   Filcfil 20782    fLimf cflf 20872
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-suc 5448  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-1o 7190  df-map 7482  df-en 7578  df-fbas 18893  df-top 19843  df-topon 19845  df-nei 20036  df-haus 20253  df-fil 20783  df-flim 20876  df-flf 20877
This theorem is referenced by:  cnextfvval  21002  cnextcn  21004  cnextfres1  21005
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