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Theorem cncls2 20276
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 20252 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> Y )
213expia 1207 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  ->  F : X
--> Y ) )
3 elpwi 3988 . . . . . . 7  |-  ( x  e.  ~P Y  ->  x  C_  Y )
43adantl 467 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_  Y )
5 toponuni 19929 . . . . . . 7  |-  ( K  e.  (TopOn `  Y
)  ->  Y  =  U. K )
65ad2antlr 731 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  Y  =  U. K )
74, 6sseqtrd 3500 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_ 
U. K )
8 eqid 2422 . . . . . . 7  |-  U. K  =  U. K
98cncls2i 20273 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  x  C_  U. K )  ->  ( ( cls `  J ) `  ( `' F " x ) )  C_  ( `' F " ( ( cls `  K ) `  x
) ) )
109expcom 436 . . . . 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 2843 . . 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 535 . 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 20032 . . . . . . . . 9  |-  ( Clsd `  K )  C_  ~P U. K
155ad2antlr 731 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  Y  =  U. K )
1615pweqd 3984 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ~P Y  =  ~P U. K
)
1714, 16syl5sseqr 3513 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ( Clsd `  K )  C_  ~P Y )
1817sseld 3463 . . . . . . 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 20044 . . . . . . . . . . . 12  |-  ( x  e.  ( Clsd `  K
)  ->  ( ( cls `  K ) `  x )  =  x )
2120ad2antll 733 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( ( cls `  K
) `  x )  =  x )
2221imaeq2d 5184 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( `' F "
( ( cls `  K
) `  x )
)  =  ( `' F " x ) )
2322sseq2d 3492 . . . . . . . . 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 19928 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2524ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  J  e.  Top )
26 cnvimass 5204 . . . . . . . . . . 11  |-  ( `' F " x ) 
C_  dom  F
27 fdm 5747 . . . . . . . . . . . . 13  |-  ( F : X --> Y  ->  dom  F  =  X )
2827ad2antrl 732 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  dom  F  =  X )
29 toponuni 19929 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3029ad2antrr 730 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  X  =  U. J )
3128, 30eqtrd 2463 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  ->  dom  F  =  U. J
)
3226, 31syl5sseq 3512 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( F : X --> Y  /\  x  e.  ( Clsd `  K
) ) )  -> 
( `' F "
x )  C_  U. J
)
33 eqid 2422 . . . . . . . . . . 11  |-  U. J  =  U. J
3433iscld4 20068 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( `' F " x ) 
C_  U. J )  -> 
( ( `' F " x )  e.  (
Clsd `  J )  <->  ( ( cls `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x ) ) )
3525, 32, 34syl2anc 665 . . . . . . . . 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 259 . . . . . . . 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 618 . . . . . . 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 250 . . . . . 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 217 . . . . 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 2832 . . . 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 697 . . 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 20272 . . 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 237 . 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 193 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 187    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775    C_ wss 3436   ~Pcpw 3979   U.cuni 4216   `'ccnv 4849   dom cdm 4850   "cima 4853   -->wf 5594   ` cfv 5598  (class class class)co 6302   Topctop 19904  TopOnctopon 19905   Clsdccld 20018   clsccl 20020    Cn ccn 20227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-map 7479  df-top 19908  df-topon 19910  df-cld 20021  df-cls 20023  df-cn 20230
This theorem is referenced by:  cncls  20277
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