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Theorem cnntr 12431
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 12411 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> Y )
213expia 1184 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  ->  F : X
--> Y ) )
3 elpwi 3523 . . . . . . 7  |-  ( x  e.  ~P Y  ->  x  C_  Y )
43adantl 275 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_  Y )
5 toponuni 12219 . . . . . . 7  |-  ( K  e.  (TopOn `  Y
)  ->  Y  =  U. K )
65ad2antlr 481 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  Y  =  U. K )
74, 6sseqtrd 3139 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_ 
U. K )
8 eqid 2140 . . . . . . 7  |-  U. K  =  U. K
98cnntri 12430 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  x  C_  U. K )  ->  ( `' F " ( ( int `  K
) `  x )
)  C_  ( ( int `  J ) `  ( `' F " x ) ) )
109expcom 115 . . . . 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 2515 . . 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 305 . 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 12230 . . . . . . . . . 10  |-  ( ( K  e.  (TopOn `  Y )  /\  x  e.  K )  ->  x  C_  Y )
15 velpw 3521 . . . . . . . . . 10  |-  ( x  e.  ~P Y  <->  x  C_  Y
)
1614, 15sylibr 133 . . . . . . . . 9  |-  ( ( K  e.  (TopOn `  Y )  /\  x  e.  K )  ->  x  e.  ~P Y )
1716ex 114 . . . . . . . 8  |-  ( K  e.  (TopOn `  Y
)  ->  ( x  e.  K  ->  x  e. 
~P Y ) )
1817ad2antlr 481 . . . . . . 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 12218 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2120ad3antrrr 484 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  J  e.  Top )
22 cnvimass 4909 . . . . . . . . . . 11  |-  ( `' F " x ) 
C_  dom  F
23 fdm 5285 . . . . . . . . . . . . 13  |-  ( F : X --> Y  ->  dom  F  =  X )
2423ad2antlr 481 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  dom  F  =  X )
25 toponuni 12219 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
2625ad3antrrr 484 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  X  =  U. J )
2724, 26eqtrd 2173 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  dom  F  = 
U. J )
2822, 27sseqtrid 3151 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( `' F " x )  C_  U. J )
29 eqid 2140 . . . . . . . . . . 11  |-  U. J  =  U. J
3029ntrss2 12327 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( `' F " x ) 
C_  U. J )  -> 
( ( int `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x ) )
3121, 28, 30syl2anc 409 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( ( int `  J ) `  ( `' F " x ) )  C_  ( `' F " x ) )
32 eqss 3116 . . . . . . . . . 10  |-  ( ( ( int `  J
) `  ( `' F " x ) )  =  ( `' F " x )  <->  ( (
( int `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x )  /\  ( `' F " x ) 
C_  ( ( int `  J ) `  ( `' F " x ) ) ) )
3332baib 905 . . . . . . . . 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 12331 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  ( `' F " x ) 
C_  U. J )  -> 
( ( `' F " x )  e.  J  <->  ( ( int `  J
) `  ( `' F " x ) )  =  ( `' F " x ) ) )
3621, 28, 35syl2anc 409 . . . . . . . 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 12218 . . . . . . . . . . . 12  |-  ( K  e.  (TopOn `  Y
)  ->  K  e.  Top )
3837ad3antlr 485 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  K  e.  Top )
39 isopn3i 12341 . . . . . . . . . . 11  |-  ( ( K  e.  Top  /\  x  e.  K )  ->  ( ( int `  K
) `  x )  =  x )
4038, 39sylancom 417 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( ( int `  K ) `  x )  =  x )
4140imaeq2d 4888 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( `' F " ( ( int `  K ) `  x
) )  =  ( `' F " x ) )
4241sseq1d 3130 . . . . . . . 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 220 . . . . . . 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 440 . . . . . 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 148 . . . . 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 2505 . . . 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 445 . . 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 12403 . . 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 168 . 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 128 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 103    <-> wb 104    = wceq 1332    e. wcel 1481   A.wral 2417    C_ wss 3075   ~Pcpw 3514   U.cuni 3743   `'ccnv 4545   dom cdm 4546   "cima 4549   -->wf 5126   ` cfv 5130  (class class class)co 5781   Topctop 12201  TopOnctopon 12214   intcnt 12299    Cn ccn 12391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-map 6551  df-top 12202  df-topon 12215  df-ntr 12302  df-cn 12394
This theorem is referenced by: (None)
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