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Theorem hausflf2 19413
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 3634 . . 3  |-  ( ( ( J  fLimf  L ) `
 F )  =/=  (/) 
<->  E. x  x  e.  ( ( J  fLimf  L ) `  F ) )
21biimpi 194 . 2  |-  ( ( ( J  fLimf  L ) `
 F )  =/=  (/)  ->  E. x  x  e.  ( ( J  fLimf  L ) `  F ) )
3 hausflf.x . . 3  |-  X  = 
U. J
43hausflf 19412 . 2  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  E* x  x  e.  (
( J  fLimf  L ) `
 F ) )
5 euen1b 7368 . . . 4  |-  ( ( ( J  fLimf  L ) `
 F )  ~~  1o 
<->  E! x  x  e.  ( ( J  fLimf  L ) `  F ) )
6 eu5 2281 . . . 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 250 . . 3  |-  ( ( E. x  x  e.  ( ( J  fLimf  L ) `  F )  /\  E* x  x  e.  ( ( J 
fLimf  L ) `  F
) )  <->  ( ( J  fLimf  L ) `  F )  ~~  1o )
87biimpi 194 . 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 475 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 369    /\ w3a 958    = wceq 1362   E.wex 1589    e. wcel 1755   E!weu 2254   E*wmo 2255    =/= wne 2596   (/)c0 3625   U.cuni 4079   class class class wbr 4280   -->wf 5402   ` cfv 5406  (class class class)co 6080   1oc1o 6901    ~~ cen 7295   Hauscha 18754   Filcfil 19260    fLimf cflf 19350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1o 6908  df-map 7204  df-en 7299  df-fbas 17658  df-top 18345  df-topon 18348  df-nei 18544  df-haus 18761  df-fil 19261  df-flim 19354  df-flf 19355
This theorem is referenced by:  cnextfvval  19479  cnextcn  19481  cnextfres  19482
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