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Theorem cnntr 17004
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 16979 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> Y )
213expia 1153 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  ->  F : X
--> Y ) )
3 elpwi 3633 . . . . . . 7  |-  ( x  e.  ~P Y  ->  x  C_  Y )
43adantl 452 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_  Y )
5 toponuni 16665 . . . . . . 7  |-  ( K  e.  (TopOn `  Y
)  ->  Y  =  U. K )
65ad2antlr 707 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  Y  =  U. K )
74, 6sseqtrd 3214 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  x  e.  ~P Y )  ->  x  C_ 
U. K )
8 eqid 2283 . . . . . . 7  |-  U. K  =  U. K
98cnntri 17000 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  x  C_  U. K )  ->  ( `' F " ( ( int `  K
) `  x )
)  C_  ( ( int `  J ) `  ( `' F " x ) ) )
109expcom 424 . . . . 5  |-  ( x 
C_  U. K  ->  ( F  e.  ( J  Cn  K )  ->  ( `' F " ( ( int `  K ) `
 x ) ) 
C_  ( ( int `  J ) `  ( `' F " x ) ) ) )
117, 10syl 15 . . . 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 2633 . . 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 519 . 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 16667 . . . . . . . . . 10  |-  ( ( K  e.  (TopOn `  Y )  /\  x  e.  K )  ->  x  C_  Y )
15 vex 2791 . . . . . . . . . . 11  |-  x  e. 
_V
1615elpw 3631 . . . . . . . . . 10  |-  ( x  e.  ~P Y  <->  x  C_  Y
)
1714, 16sylibr 203 . . . . . . . . 9  |-  ( ( K  e.  (TopOn `  Y )  /\  x  e.  K )  ->  x  e.  ~P Y )
1817ex 423 . . . . . . . 8  |-  ( K  e.  (TopOn `  Y
)  ->  ( x  e.  K  ->  x  e. 
~P Y ) )
1918ad2antlr 707 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  (
x  e.  K  ->  x  e.  ~P Y
) )
2019imim1d 69 . . . . . 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 ) ) ) ) )
21 topontop 16664 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2221ad3antrrr 710 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  J  e.  Top )
23 cnvimass 5033 . . . . . . . . . . 11  |-  ( `' F " x ) 
C_  dom  F
24 fdm 5393 . . . . . . . . . . . . 13  |-  ( F : X --> Y  ->  dom  F  =  X )
2524ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  dom  F  =  X )
26 toponuni 16665 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
2726ad3antrrr 710 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  X  =  U. J )
2825, 27eqtrd 2315 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  dom  F  = 
U. J )
2923, 28syl5sseq 3226 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( `' F " x )  C_  U. J )
30 eqid 2283 . . . . . . . . . . 11  |-  U. J  =  U. J
3130ntrss2 16794 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( `' F " x ) 
C_  U. J )  -> 
( ( int `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x ) )
3222, 29, 31syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( ( int `  J ) `  ( `' F " x ) )  C_  ( `' F " x ) )
33 eqss 3194 . . . . . . . . . 10  |-  ( ( ( int `  J
) `  ( `' F " x ) )  =  ( `' F " x )  <->  ( (
( int `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x )  /\  ( `' F " x ) 
C_  ( ( int `  J ) `  ( `' F " x ) ) ) )
3433baib 871 . . . . . . . . 9  |-  ( ( ( int `  J
) `  ( `' F " x ) ) 
C_  ( `' F " x )  ->  (
( ( int `  J
) `  ( `' F " x ) )  =  ( `' F " x )  <->  ( `' F " x )  C_  ( ( int `  J
) `  ( `' F " x ) ) ) )
3532, 34syl 15 . . . . . . . 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 ) ) ) )
3630isopn3 16803 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  ( `' F " x ) 
C_  U. J )  -> 
( ( `' F " x )  e.  J  <->  ( ( int `  J
) `  ( `' F " x ) )  =  ( `' F " x ) ) )
3722, 29, 36syl2anc 642 . . . . . . . 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 ) ) )
38 simpllr 735 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  K  e.  (TopOn `  Y ) )
39 topontop 16664 . . . . . . . . . . . 12  |-  ( K  e.  (TopOn `  Y
)  ->  K  e.  Top )
4038, 39syl 15 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  K  e.  Top )
41 isopn3i 16819 . . . . . . . . . . 11  |-  ( ( K  e.  Top  /\  x  e.  K )  ->  ( ( int `  K
) `  x )  =  x )
4240, 41sylancom 648 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( ( int `  K ) `  x )  =  x )
4342imaeq2d 5012 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  x  e.  K
)  ->  ( `' F " ( ( int `  K ) `  x
) )  =  ( `' F " x ) )
4443sseq1d 3205 . . . . . . . 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 ) ) ) )
4535, 37, 443bitr4rd 277 . . . . . . 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
) )
4645pm5.74da 668 . . . . . 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 ) ) )
4720, 46sylibd 205 . . . . 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
) ) )
4847ralimdv2 2623 . . . 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 ) )
4948imdistanda 674 . . 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 ) ) )
50 iscn 16965 . . 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 ) ) )
5149, 50sylibrd 225 . 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
) ) )
5213, 51impbid 183 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    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   `'ccnv 4688   dom cdm 4689   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   Topctop 16631  TopOnctopon 16632   intcnt 16754    Cn ccn 16954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-top 16636  df-topon 16639  df-ntr 16757  df-cn 16957
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