MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hausflf2 Structured version   Unicode version

Theorem hausflf2 19689
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 3746 . . 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 19688 . 2  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  E* x  x  e.  (
( J  fLimf  L ) `
 F ) )
5 euen1b 7482 . . . 4  |-  ( ( ( J  fLimf  L ) `
 F )  ~~  1o 
<->  E! x  x  e.  ( ( J  fLimf  L ) `  F ) )
6 eu5 2290 . . . 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 478 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 965    = wceq 1370   E.wex 1587    e. wcel 1758   E!weu 2260   E*wmo 2261    =/= wne 2644   (/)c0 3737   U.cuni 4191   class class class wbr 4392   -->wf 5514   ` cfv 5518  (class class class)co 6192   1oc1o 7015    ~~ cen 7409   Hauscha 19030   Filcfil 19536    fLimf cflf 19626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1o 7022  df-map 7318  df-en 7413  df-fbas 17925  df-top 18621  df-topon 18624  df-nei 18820  df-haus 19037  df-fil 19537  df-flim 19630  df-flf 19631
This theorem is referenced by:  cnextfvval  19755  cnextcn  19757  cnextfres  19758
  Copyright terms: Public domain W3C validator