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Theorem cnflf2 17698
Description: A function is continuous iff it respects filter limits. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
cnflf2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. f  e.  ( Fil `  X
) ( F "
( J  fLim  f
) )  C_  (
( K  fLimf  f ) `
 F ) ) ) )
Distinct variable groups:    f, X    f, Y    f, F    f, J    f, K

Proof of Theorem cnflf2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cnflf 17697 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. f  e.  ( Fil `  X
) A. x  e.  ( J  fLim  f
) ( F `  x )  e.  ( ( K  fLimf  f ) `
 F ) ) ) )
2 ffun 5391 . . . . . 6  |-  ( F : X --> Y  ->  Fun  F )
32adantl 452 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  Fun  F )
4 eqid 2283 . . . . . . . 8  |-  U. J  =  U. J
54flimelbas 17663 . . . . . . 7  |-  ( x  e.  ( J  fLim  f )  ->  x  e.  U. J )
65ssriv 3184 . . . . . 6  |-  ( J 
fLim  f )  C_  U. J
7 fdm 5393 . . . . . . . 8  |-  ( F : X --> Y  ->  dom  F  =  X )
87adantl 452 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  dom  F  =  X )
9 toponuni 16665 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
109ad2antrr 706 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  X  =  U. J )
118, 10eqtrd 2315 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  dom  F  =  U. J )
126, 11syl5sseqr 3227 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ( J  fLim  f )  C_  dom  F )
13 funimass4 5573 . . . . 5  |-  ( ( Fun  F  /\  ( J  fLim  f )  C_  dom  F )  ->  (
( F " ( J  fLim  f ) ) 
C_  ( ( K 
fLimf  f ) `  F
)  <->  A. x  e.  ( J  fLim  f )
( F `  x
)  e.  ( ( K  fLimf  f ) `  F ) ) )
143, 12, 13syl2anc 642 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  (
( F " ( J  fLim  f ) ) 
C_  ( ( K 
fLimf  f ) `  F
)  <->  A. x  e.  ( J  fLim  f )
( F `  x
)  e.  ( ( K  fLimf  f ) `  F ) ) )
1514ralbidv 2563 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ( A. f  e.  ( Fil `  X ) ( F " ( J 
fLim  f ) ) 
C_  ( ( K 
fLimf  f ) `  F
)  <->  A. f  e.  ( Fil `  X ) A. x  e.  ( J  fLim  f )
( F `  x
)  e.  ( ( K  fLimf  f ) `  F ) ) )
1615pm5.32da 622 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( ( F : X --> Y  /\  A. f  e.  ( Fil `  X ) ( F
" ( J  fLim  f ) )  C_  (
( K  fLimf  f ) `
 F ) )  <-> 
( F : X --> Y  /\  A. f  e.  ( Fil `  X
) A. x  e.  ( J  fLim  f
) ( F `  x )  e.  ( ( K  fLimf  f ) `
 F ) ) ) )
171, 16bitr4d 247 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. f  e.  ( Fil `  X
) ( F "
( J  fLim  f
) )  C_  (
( K  fLimf  f ) `
 F ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   U.cuni 3827   dom cdm 4689   "cima 4692   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858  TopOnctopon 16632    Cn ccn 16954   Filcfil 17540    fLim cflim 17629    fLimf cflf 17630
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-1st 6122  df-2nd 6123  df-map 6774  df-topgen 13344  df-top 16636  df-topon 16639  df-ntr 16757  df-nei 16835  df-cn 16957  df-cnp 16958  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635
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