ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnntr Unicode version

Theorem cnntr 13996
Description: Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
cnntr  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. x  e.  ~P  Y ( `' F " ( ( int `  K ) `
 x ) ) 
C_  ( ( int `  J ) `  ( `' F " x ) ) ) ) )
Distinct variable groups:    x, F    x, J    x, K    x, X    x, Y

Proof of Theorem cnntr
StepHypRef Expression
1 cnf2 13976 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> Y )
213expia 1206 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  ->  F : X
--> Y ) )
3 elpwi 3596 . . . . . . 7  |-  ( x  e.  ~P Y  ->  x  C_  Y )
43adantl 277 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_  Y )
5 toponuni 13786 . . . . . . 7  |-  ( K  e.  (TopOn `  Y
)  ->  Y  =  U. K )
65ad2antlr 489 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  Y  =  U. K )
74, 6sseqtrd 3205 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_ 
U. K )
8 eqid 2187 . . . . . . 7  |-  U. K  =  U. K
98cnntri 13995 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  x  C_  U. K )  ->  ( `' F " ( ( int `  K
) `  x )
)  C_  ( ( int `  J ) `  ( `' F " x ) ) )
109expcom 116 . . . . 5  |-  ( x 
C_  U. K  ->  ( F  e.  ( J  Cn  K )  ->  ( `' F " ( ( int `  K ) `
 x ) ) 
C_  ( ( int `  J ) `  ( `' F " x ) ) ) )
117, 10syl 14 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  ( F  e.  ( J  Cn  K )  ->  ( `' F " ( ( int `  K ) `
 x ) ) 
C_  ( ( int `  J ) `  ( `' F " x ) ) ) )
1211ralrimdva 2567 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  ->  A. x  e.  ~P  Y ( `' F " ( ( int `  K ) `
 x ) ) 
C_  ( ( int `  J ) `  ( `' F " x ) ) ) )
132, 12jcad 307 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  ->  ( F : X --> Y  /\  A. x  e.  ~P  Y
( `' F "
( ( int `  K
) `  x )
)  C_  ( ( int `  J ) `  ( `' F " x ) ) ) ) )
14 toponss 13797 . . . . . . . . . 10  |-  ( ( K  e.  (TopOn `  Y )  /\  x  e.  K )  ->  x  C_  Y )
15 velpw 3594 . . . . . . . . . 10  |-  ( x  e.  ~P Y  <->  x  C_  Y
)
1614, 15sylibr 134 . . . . . . . . 9  |-  ( ( K  e.  (TopOn `  Y )  /\  x  e.  K )  ->  x  e.  ~P Y )
1716ex 115 . . . . . . . 8  |-  ( K  e.  (TopOn `  Y
)  ->  ( x  e.  K  ->  x  e. 
~P Y ) )
1817ad2antlr 489 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  (
x  e.  K  ->  x  e.  ~P Y
) )
1918imim1d 75 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  (
( x  e.  ~P Y  ->  ( `' F " ( ( int `  K
) `  x )
)  C_  ( ( int `  J ) `  ( `' F " x ) ) )  ->  (
x  e.  K  -> 
( `' F "
( ( int `  K
) `  x )
)  C_  ( ( int `  J ) `  ( `' F " x ) ) ) ) )
20 topontop 13785 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2120ad3antrrr 492 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  J  e.  Top )
22 cnvimass 5003 . . . . . . . . . . 11  |-  ( `' F " x ) 
C_  dom  F
23 fdm 5383 . . . . . . . . . . . . 13  |-  ( F : X --> Y  ->  dom  F  =  X )
2423ad2antlr 489 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  dom  F  =  X )
25 toponuni 13786 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
2625ad3antrrr 492 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  X  =  U. J )
2724, 26eqtrd 2220 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  dom  F  = 
U. J )
2822, 27sseqtrid 3217 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( `' F " x )  C_  U. J )
29 eqid 2187 . . . . . . . . . . 11  |-  U. J  =  U. J
3029ntrss2 13892 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( `' F " x ) 
C_  U. J )  -> 
( ( int `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x ) )
3121, 28, 30syl2anc 411 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( ( int `  J ) `  ( `' F " x ) )  C_  ( `' F " x ) )
32 eqss 3182 . . . . . . . . . 10  |-  ( ( ( int `  J
) `  ( `' F " x ) )  =  ( `' F " x )  <->  ( (
( int `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x )  /\  ( `' F " x ) 
C_  ( ( int `  J ) `  ( `' F " x ) ) ) )
3332baib 920 . . . . . . . . 9  |-  ( ( ( int `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x )  ->  (
( ( int `  J
) `  ( `' F " x ) )  =  ( `' F " x )  <->  ( `' F " x )  C_  ( ( int `  J
) `  ( `' F " x ) ) ) )
3431, 33syl 14 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( (
( int `  J
) `  ( `' F " x ) )  =  ( `' F " x )  <->  ( `' F " x )  C_  ( ( int `  J
) `  ( `' F " x ) ) ) )
3529isopn3 13896 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  ( `' F " x ) 
C_  U. J )  -> 
( ( `' F " x )  e.  J  <->  ( ( int `  J
) `  ( `' F " x ) )  =  ( `' F " x ) ) )
3621, 28, 35syl2anc 411 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( ( `' F " x )  e.  J  <->  ( ( int `  J ) `  ( `' F " x ) )  =  ( `' F " x ) ) )
37 topontop 13785 . . . . . . . . . . . 12  |-  ( K  e.  (TopOn `  Y
)  ->  K  e.  Top )
3837ad3antlr 493 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  K  e.  Top )
39 isopn3i 13906 . . . . . . . . . . 11  |-  ( ( K  e.  Top  /\  x  e.  K )  ->  ( ( int `  K
) `  x )  =  x )
4038, 39sylancom 420 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( ( int `  K ) `  x )  =  x )
4140imaeq2d 4982 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( `' F " ( ( int `  K ) `  x
) )  =  ( `' F " x ) )
4241sseq1d 3196 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( ( `' F " ( ( int `  K ) `
 x ) ) 
C_  ( ( int `  J ) `  ( `' F " x ) )  <->  ( `' F " x )  C_  (
( int `  J
) `  ( `' F " x ) ) ) )
4334, 36, 423bitr4rd 221 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( ( `' F " ( ( int `  K ) `
 x ) ) 
C_  ( ( int `  J ) `  ( `' F " x ) )  <->  ( `' F " x )  e.  J
) )
4443pm5.74da 443 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  (
( x  e.  K  ->  ( `' F "
( ( int `  K
) `  x )
)  C_  ( ( int `  J ) `  ( `' F " x ) ) )  <->  ( x  e.  K  ->  ( `' F " x )  e.  J ) ) )
4519, 44sylibd 149 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  (
( x  e.  ~P Y  ->  ( `' F " ( ( int `  K
) `  x )
)  C_  ( ( int `  J ) `  ( `' F " x ) ) )  ->  (
x  e.  K  -> 
( `' F "
x )  e.  J
) ) )
4645ralimdv2 2557 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ( A. x  e.  ~P  Y ( `' F " ( ( int `  K
) `  x )
)  C_  ( ( int `  J ) `  ( `' F " x ) )  ->  A. x  e.  K  ( `' F " x )  e.  J ) )
4746imdistanda 448 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( ( F : X --> Y  /\  A. x  e.  ~P  Y
( `' F "
( ( int `  K
) `  x )
)  C_  ( ( int `  J ) `  ( `' F " x ) ) )  ->  ( F : X --> Y  /\  A. x  e.  K  ( `' F " x )  e.  J ) ) )
48 iscn 13968 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. x  e.  K  ( `' F " x )  e.  J ) ) )
4947, 48sylibrd 169 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( ( F : X --> Y  /\  A. x  e.  ~P  Y
( `' F "
( ( int `  K
) `  x )
)  C_  ( ( int `  J ) `  ( `' F " x ) ) )  ->  F  e.  ( J  Cn  K
) ) )
5013, 49impbid 129 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. x  e.  ~P  Y ( `' F " ( ( int `  K ) `
 x ) ) 
C_  ( ( int `  J ) `  ( `' F " x ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1363    e. wcel 2158   A.wral 2465    C_ wss 3141   ~Pcpw 3587   U.cuni 3821   `'ccnv 4637   dom cdm 4638   "cima 4641   -->wf 5224   ` cfv 5228  (class class class)co 5888   Topctop 13768  TopOnctopon 13781   intcnt 13864    Cn ccn 13956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-map 6663  df-top 13769  df-topon 13782  df-ntr 13867  df-cn 13959
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator