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Theorem cncls2 20289
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 20265 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> Y )
213expia 1210 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  ->  F : X
--> Y ) )
3 elpwi 3960 . . . . . . 7  |-  ( x  e.  ~P Y  ->  x  C_  Y )
43adantl 468 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_  Y )
5 toponuni 19942 . . . . . . 7  |-  ( K  e.  (TopOn `  Y
)  ->  Y  =  U. K )
65ad2antlr 733 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  Y  =  U. K )
74, 6sseqtrd 3468 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_ 
U. K )
8 eqid 2451 . . . . . . 7  |-  U. K  =  U. K
98cncls2i 20286 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  x  C_  U. K )  ->  ( ( cls `  J ) `  ( `' F " x ) )  C_  ( `' F " ( ( cls `  K ) `  x
) ) )
109expcom 437 . . . . 5  |-  ( x 
C_  U. K  ->  ( F  e.  ( J  Cn  K )  ->  (
( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " ( ( cls `  K
) `  x )
) ) )
117, 10syl 17 . . . 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 2806 . . 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 536 . 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 20045 . . . . . . . . 9  |-  ( Clsd `  K )  C_  ~P U. K
155ad2antlr 733 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  Y  =  U. K )
1615pweqd 3956 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ~P Y  =  ~P U. K
)
1714, 16syl5sseqr 3481 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ( Clsd `  K )  C_  ~P Y )
1817sseld 3431 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  (
x  e.  ( Clsd `  K )  ->  x  e.  ~P Y ) )
1918imim1d 78 . . . . . 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 20057 . . . . . . . . . . . 12  |-  ( x  e.  ( Clsd `  K
)  ->  ( ( cls `  K ) `  x )  =  x )
2120ad2antll 735 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( ( cls `  K
) `  x )  =  x )
2221imaeq2d 5168 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( `' F "
( ( cls `  K
) `  x )
)  =  ( `' F " x ) )
2322sseq2d 3460 . . . . . . . . 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 19941 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2524ad2antrr 732 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  J  e.  Top )
26 cnvimass 5188 . . . . . . . . . . 11  |-  ( `' F " x ) 
C_  dom  F
27 fdm 5733 . . . . . . . . . . . . 13  |-  ( F : X --> Y  ->  dom  F  =  X )
2827ad2antrl 734 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  dom  F  =  X )
29 toponuni 19942 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3029ad2antrr 732 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  X  =  U. J )
3128, 30eqtrd 2485 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  dom  F  =  U. J
)
3226, 31syl5sseq 3480 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( `' F "
x )  C_  U. J
)
33 eqid 2451 . . . . . . . . . . 11  |-  U. J  =  U. J
3433iscld4 20081 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( `' F " x ) 
C_  U. J )  -> 
( ( `' F " x )  e.  (
Clsd `  J )  <->  ( ( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x ) ) )
3525, 32, 34syl2anc 667 . . . . . . . . 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 260 . . . . . . . 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 620 . . . . . . 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 251 . . . . . 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 218 . . . . 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 2795 . . . 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 699 . . 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 20285 . . 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 238 . 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 194 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 setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737    C_ wss 3404   ~Pcpw 3951   U.cuni 4198   `'ccnv 4833   dom cdm 4834   "cima 4837   -->wf 5578   ` cfv 5582  (class class class)co 6290   Topctop 19917  TopOnctopon 19918   Clsdccld 20031   clsccl 20033    Cn ccn 20240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-map 7474  df-top 19921  df-topon 19923  df-cld 20034  df-cls 20036  df-cn 20243
This theorem is referenced by:  cncls  20290
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