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Theorem cncls2 17339
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 17315 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> Y )
213expia 1156 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  ->  F : X
--> Y ) )
3 elpwi 3809 . . . . . . 7  |-  ( x  e.  ~P Y  ->  x  C_  Y )
43adantl 454 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_  Y )
5 toponuni 16994 . . . . . . 7  |-  ( K  e.  (TopOn `  Y
)  ->  Y  =  U. K )
65ad2antlr 709 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  Y  =  U. K )
74, 6sseqtrd 3386 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_ 
U. K )
8 eqid 2438 . . . . . . 7  |-  U. K  =  U. K
98cncls2i 17336 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  x  C_  U. K )  ->  ( ( cls `  J ) `  ( `' F " x ) )  C_  ( `' F " ( ( cls `  K ) `  x
) ) )
109expcom 426 . . . . 5  |-  ( x 
C_  U. K  ->  ( F  e.  ( J  Cn  K )  ->  (
( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " ( ( cls `  K
) `  x )
) ) )
117, 10syl 16 . . . 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 2798 . . 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 521 . 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 17096 . . . . . . . . 9  |-  ( Clsd `  K )  C_  ~P U. K
155ad2antlr 709 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  Y  =  U. K )
1615pweqd 3806 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ~P Y  =  ~P U. K
)
1714, 16syl5sseqr 3399 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ( Clsd `  K )  C_  ~P Y )
1817sseld 3349 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  (
x  e.  ( Clsd `  K )  ->  x  e.  ~P Y ) )
1918imim1d 72 . . . . . 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 17108 . . . . . . . . . . . 12  |-  ( x  e.  ( Clsd `  K
)  ->  ( ( cls `  K ) `  x )  =  x )
2120ad2antll 711 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( ( cls `  K
) `  x )  =  x )
2221imaeq2d 5205 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( `' F "
( ( cls `  K
) `  x )
)  =  ( `' F " x ) )
2322sseq2d 3378 . . . . . . . . 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 16993 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2524ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  J  e.  Top )
26 cnvimass 5226 . . . . . . . . . . 11  |-  ( `' F " x ) 
C_  dom  F
27 fdm 5597 . . . . . . . . . . . . 13  |-  ( F : X --> Y  ->  dom  F  =  X )
2827ad2antrl 710 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  dom  F  =  X )
29 toponuni 16994 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3029ad2antrr 708 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  X  =  U. J )
3128, 30eqtrd 2470 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  dom  F  =  U. J
)
3226, 31syl5sseq 3398 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( `' F "
x )  C_  U. J
)
33 eqid 2438 . . . . . . . . . . 11  |-  U. J  =  U. J
3433iscld4 17131 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( `' F " x ) 
C_  U. J )  -> 
( ( `' F " x )  e.  (
Clsd `  J )  <->  ( ( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x ) ) )
3525, 32, 34syl2anc 644 . . . . . . . . 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 249 . . . . . . . 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 600 . . . . . . 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 240 . . . . . 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 207 . . . . 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 2788 . . . 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 676 . . 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 17335 . . 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 227 . 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 185 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   `'ccnv 4879   dom cdm 4880   "cima 4883   -->wf 5452   ` cfv 5456  (class class class)co 6083   Topctop 16960  TopOnctopon 16961   Clsdccld 17082   clsccl 17084    Cn ccn 17290
This theorem is referenced by:  cncls  17340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-top 16965  df-topon 16968  df-cld 17085  df-cls 17087  df-cn 17293
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