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Theorem cnflf2 17750
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 17749 . 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 5429 . . . . . 6  |-  ( F : X --> Y  ->  Fun  F )
32adantl 452 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  Fun  F )
4 eqid 2316 . . . . . . . 8  |-  U. J  =  U. J
54flimelbas 17715 . . . . . . 7  |-  ( x  e.  ( J  fLim  f )  ->  x  e.  U. J )
65ssriv 3218 . . . . . 6  |-  ( J 
fLim  f )  C_  U. J
7 fdm 5431 . . . . . . . 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 16721 . . . . . . . 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 2348 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  dom  F  =  U. J )
126, 11syl5sseqr 3261 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ( J  fLim  f )  C_  dom  F )
13 funimass4 5611 . . . . 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 2597 . . 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 1633    e. wcel 1701   A.wral 2577    C_ wss 3186   U.cuni 3864   dom cdm 4726   "cima 4729   Fun wfun 5286   -->wf 5288   ` cfv 5292  (class class class)co 5900  TopOnctopon 16688    Cn ccn 17010   Filcfil 17592    fLim cflim 17681    fLimf cflf 17682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-map 6817  df-topgen 13393  df-fbas 16429  df-fg 16430  df-top 16692  df-topon 16695  df-ntr 16813  df-nei 16891  df-cn 17013  df-cnp 17014  df-fil 17593  df-fm 17685  df-flim 17686  df-flf 17687
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