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Theorem cnntr 20368
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 20342 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> Y )
213expia 1233 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  ->  F : X
--> Y ) )
3 elpwi 3951 . . . . . . 7  |-  ( x  e.  ~P Y  ->  x  C_  Y )
43adantl 473 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_  Y )
5 toponuni 20019 . . . . . . 7  |-  ( K  e.  (TopOn `  Y
)  ->  Y  =  U. K )
65ad2antlr 741 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  Y  =  U. K )
74, 6sseqtrd 3454 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_ 
U. K )
8 eqid 2471 . . . . . . 7  |-  U. K  =  U. K
98cnntri 20364 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  x  C_  U. K )  ->  ( `' F " ( ( int `  K
) `  x )
)  C_  ( ( int `  J ) `  ( `' F " x ) ) )
109expcom 442 . . . . 5  |-  ( x 
C_  U. K  ->  ( F  e.  ( J  Cn  K )  ->  ( `' F " ( ( int `  K ) `
 x ) ) 
C_  ( ( int `  J ) `  ( `' F " x ) ) ) )
117, 10syl 17 . . . 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 2812 . . 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 542 . 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 20021 . . . . . . . . . 10  |-  ( ( K  e.  (TopOn `  Y )  /\  x  e.  K )  ->  x  C_  Y )
15 selpw 3949 . . . . . . . . . 10  |-  ( x  e.  ~P Y  <->  x  C_  Y
)
1614, 15sylibr 217 . . . . . . . . 9  |-  ( ( K  e.  (TopOn `  Y )  /\  x  e.  K )  ->  x  e.  ~P Y )
1716ex 441 . . . . . . . 8  |-  ( K  e.  (TopOn `  Y
)  ->  ( x  e.  K  ->  x  e. 
~P Y ) )
1817ad2antlr 741 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  (
x  e.  K  ->  x  e.  ~P Y
) )
1918imim1d 77 . . . . . 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 20018 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2120ad3antrrr 744 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  J  e.  Top )
22 cnvimass 5194 . . . . . . . . . . 11  |-  ( `' F " x ) 
C_  dom  F
23 fdm 5745 . . . . . . . . . . . . 13  |-  ( F : X --> Y  ->  dom  F  =  X )
2423ad2antlr 741 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  dom  F  =  X )
25 toponuni 20019 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
2625ad3antrrr 744 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  X  =  U. J )
2724, 26eqtrd 2505 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  dom  F  = 
U. J )
2822, 27syl5sseq 3466 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( `' F " x )  C_  U. J )
29 eqid 2471 . . . . . . . . . . 11  |-  U. J  =  U. J
3029ntrss2 20149 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( `' F " x ) 
C_  U. J )  -> 
( ( int `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x ) )
3121, 28, 30syl2anc 673 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( ( int `  J ) `  ( `' F " x ) )  C_  ( `' F " x ) )
32 eqss 3433 . . . . . . . . . 10  |-  ( ( ( int `  J
) `  ( `' F " x ) )  =  ( `' F " x )  <->  ( (
( int `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x )  /\  ( `' F " x ) 
C_  ( ( int `  J ) `  ( `' F " x ) ) ) )
3332baib 919 . . . . . . . . 9  |-  ( ( ( int `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x )  ->  (
( ( int `  J
) `  ( `' F " x ) )  =  ( `' F " x )  <->  ( `' F " x )  C_  ( ( int `  J
) `  ( `' F " x ) ) ) )
3431, 33syl 17 . . . . . . . 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 20159 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  ( `' F " x ) 
C_  U. J )  -> 
( ( `' F " x )  e.  J  <->  ( ( int `  J
) `  ( `' F " x ) )  =  ( `' F " x ) ) )
3621, 28, 35syl2anc 673 . . . . . . . 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 20018 . . . . . . . . . . . 12  |-  ( K  e.  (TopOn `  Y
)  ->  K  e.  Top )
3837ad3antlr 745 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  K  e.  Top )
39 isopn3i 20175 . . . . . . . . . . 11  |-  ( ( K  e.  Top  /\  x  e.  K )  ->  ( ( int `  K
) `  x )  =  x )
4038, 39sylancom 680 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( ( int `  K ) `  x )  =  x )
4140imaeq2d 5174 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( `' F " ( ( int `  K ) `  x
) )  =  ( `' F " x ) )
4241sseq1d 3445 . . . . . . . 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 294 . . . . . . 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 701 . . . . . 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 222 . . . . 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 2804 . . . 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 707 . . 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 20328 . . 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 242 . 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 195 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 setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756    C_ wss 3390   ~Pcpw 3942   U.cuni 4190   `'ccnv 4838   dom cdm 4839   "cima 4842   -->wf 5585   ` cfv 5589  (class class class)co 6308   Topctop 19994  TopOnctopon 19995   intcnt 20109    Cn ccn 20317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-map 7492  df-top 19998  df-topon 20000  df-ntr 20112  df-cn 20320
This theorem is referenced by: (None)
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