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Theorem cncls2 17002
Description: Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
cncls2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. x  e.  ~P  Y ( ( cls `  J ) `
 ( `' F " x ) )  C_  ( `' F " ( ( cls `  K ) `
 x ) ) ) ) )
Distinct variable groups:    x, F    x, J    x, K    x, X    x, Y

Proof of Theorem cncls2
StepHypRef Expression
1 cnf2 16979 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> Y )
213expia 1153 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  ->  F : X
--> Y ) )
3 elpwi 3633 . . . . . . 7  |-  ( x  e.  ~P Y  ->  x  C_  Y )
43adantl 452 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_  Y )
5 toponuni 16665 . . . . . . 7  |-  ( K  e.  (TopOn `  Y
)  ->  Y  =  U. K )
65ad2antlr 707 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  Y  =  U. K )
74, 6sseqtrd 3214 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_ 
U. K )
8 eqid 2283 . . . . . . 7  |-  U. K  =  U. K
98cncls2i 16999 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  x  C_  U. K )  ->  ( ( cls `  J ) `  ( `' F " x ) )  C_  ( `' F " ( ( cls `  K ) `  x
) ) )
109expcom 424 . . . . 5  |-  ( x 
C_  U. K  ->  ( F  e.  ( J  Cn  K )  ->  (
( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " ( ( cls `  K
) `  x )
) ) )
117, 10syl 15 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  ( F  e.  ( J  Cn  K )  ->  (
( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " ( ( cls `  K
) `  x )
) ) )
1211ralrimdva 2633 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  ->  A. x  e.  ~P  Y ( ( cls `  J ) `
 ( `' F " x ) )  C_  ( `' F " ( ( cls `  K ) `
 x ) ) ) )
132, 12jcad 519 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  ->  ( F : X --> Y  /\  A. x  e.  ~P  Y
( ( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " ( ( cls `  K
) `  x )
) ) ) )
148cldss2 16767 . . . . . . . . 9  |-  ( Clsd `  K )  C_  ~P U. K
155ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  Y  =  U. K )
1615pweqd 3630 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ~P Y  =  ~P U. K
)
1714, 16syl5sseqr 3227 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ( Clsd `  K )  C_  ~P Y )
1817sseld 3179 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  (
x  e.  ( Clsd `  K )  ->  x  e.  ~P Y ) )
1918imim1d 69 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  (
( x  e.  ~P Y  ->  ( ( cls `  J ) `  ( `' F " x ) )  C_  ( `' F " ( ( cls `  K ) `  x
) ) )  -> 
( x  e.  (
Clsd `  K )  ->  ( ( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " ( ( cls `  K
) `  x )
) ) ) )
20 cldcls 16779 . . . . . . . . . . . 12  |-  ( x  e.  ( Clsd `  K
)  ->  ( ( cls `  K ) `  x )  =  x )
2120ad2antll 709 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( ( cls `  K
) `  x )  =  x )
2221imaeq2d 5012 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( `' F "
( ( cls `  K
) `  x )
)  =  ( `' F " x ) )
2322sseq2d 3206 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( ( ( cls `  J ) `  ( `' F " x ) )  C_  ( `' F " ( ( cls `  K ) `  x
) )  <->  ( ( cls `  J ) `  ( `' F " x ) )  C_  ( `' F " x ) ) )
24 topontop 16664 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2524ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  J  e.  Top )
26 cnvimass 5033 . . . . . . . . . . 11  |-  ( `' F " x ) 
C_  dom  F
27 fdm 5393 . . . . . . . . . . . . 13  |-  ( F : X --> Y  ->  dom  F  =  X )
2827ad2antrl 708 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  dom  F  =  X )
29 toponuni 16665 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3029ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  X  =  U. J )
3128, 30eqtrd 2315 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  dom  F  =  U. J
)
3226, 31syl5sseq 3226 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( `' F "
x )  C_  U. J
)
33 eqid 2283 . . . . . . . . . . 11  |-  U. J  =  U. J
3433iscld4 16802 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( `' F " x ) 
C_  U. J )  -> 
( ( `' F " x )  e.  (
Clsd `  J )  <->  ( ( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x ) ) )
3525, 32, 34syl2anc 642 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( ( `' F " x )  e.  (
Clsd `  J )  <->  ( ( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x ) ) )
3623, 35bitr4d 247 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( ( ( cls `  J ) `  ( `' F " x ) )  C_  ( `' F " ( ( cls `  K ) `  x
) )  <->  ( `' F " x )  e.  ( Clsd `  J
) ) )
3736expr 598 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  (
x  e.  ( Clsd `  K )  ->  (
( ( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " ( ( cls `  K
) `  x )
)  <->  ( `' F " x )  e.  (
Clsd `  J )
) ) )
3837pm5.74d 238 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  (
( x  e.  (
Clsd `  K )  ->  ( ( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " ( ( cls `  K
) `  x )
) )  <->  ( x  e.  ( Clsd `  K
)  ->  ( `' F " x )  e.  ( Clsd `  J
) ) ) )
3919, 38sylibd 205 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  (
( x  e.  ~P Y  ->  ( ( cls `  J ) `  ( `' F " x ) )  C_  ( `' F " ( ( cls `  K ) `  x
) ) )  -> 
( x  e.  (
Clsd `  K )  ->  ( `' F "
x )  e.  (
Clsd `  J )
) ) )
4039ralimdv2 2623 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ( A. x  e.  ~P  Y ( ( cls `  J ) `  ( `' F " x ) )  C_  ( `' F " ( ( cls `  K ) `  x
) )  ->  A. x  e.  ( Clsd `  K
) ( `' F " x )  e.  (
Clsd `  J )
) )
4140imdistanda 674 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( ( F : X --> Y  /\  A. x  e.  ~P  Y
( ( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " ( ( cls `  K
) `  x )
) )  ->  ( F : X --> Y  /\  A. x  e.  ( Clsd `  K ) ( `' F " x )  e.  ( Clsd `  J
) ) ) )
42 iscncl 16998 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. x  e.  ( Clsd `  K
) ( `' F " x )  e.  (
Clsd `  J )
) ) )
4341, 42sylibrd 225 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( ( F : X --> Y  /\  A. x  e.  ~P  Y
( ( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " ( ( cls `  K
) `  x )
) )  ->  F  e.  ( J  Cn  K
) ) )
4413, 43impbid 183 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. x  e.  ~P  Y ( ( cls `  J ) `
 ( `' F " x ) )  C_  ( `' F " ( ( cls `  K ) `
 x ) ) ) ) )
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   ~Pcpw 3625   U.cuni 3827   `'ccnv 4688   dom cdm 4689   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   Topctop 16631  TopOnctopon 16632   Clsdccld 16753   clsccl 16755    Cn ccn 16954
This theorem is referenced by:  cncls  17003
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-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-int 3863  df-iun 3907  df-iin 3908  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-map 6774  df-top 16636  df-topon 16639  df-cld 16756  df-cls 16758  df-cn 16957
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