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Theorem cnflf2 21016
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 21015 . 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 5748 . . . . . 6  |-  ( F : X --> Y  ->  Fun  F )
32adantl 467 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  Fun  F )
4 eqid 2422 . . . . . . . 8  |-  U. J  =  U. J
54flimelbas 20981 . . . . . . 7  |-  ( x  e.  ( J  fLim  f )  ->  x  e.  U. J )
65ssriv 3468 . . . . . 6  |-  ( J 
fLim  f )  C_  U. J
7 fdm 5750 . . . . . . . 8  |-  ( F : X --> Y  ->  dom  F  =  X )
87adantl 467 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  dom  F  =  X )
9 toponuni 19940 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
109ad2antrr 730 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  X  =  U. J )
118, 10eqtrd 2463 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  dom  F  =  U. J )
126, 11syl5sseqr 3513 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ( J  fLim  f )  C_  dom  F )
13 funimass4 5932 . . . . 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 665 . . . 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 2861 . . 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 645 . 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 259 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771    C_ wss 3436   U.cuni 4219   dom cdm 4853   "cima 4856   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6305  TopOnctopon 19916    Cn ccn 20238   Filcfil 20858    fLim cflim 20947    fLimf cflf 20948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  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 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  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-1st 6807  df-2nd 6808  df-map 7485  df-topgen 15341  df-fbas 18966  df-fg 18967  df-top 19919  df-topon 19921  df-ntr 20033  df-nei 20112  df-cn 20241  df-cnp 20242  df-fil 20859  df-fm 20951  df-flim 20952  df-flf 20953
This theorem is referenced by: (None)
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