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Theorem opncldf1 16837
Description: A bijection useful for converting statements about open sets to statements about closed sets and vice versa. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
opncldf.1  |-  X  = 
U. J
opncldf.2  |-  F  =  ( u  e.  J  |->  ( X  \  u
) )
Assertion
Ref Expression
opncldf1  |-  ( J  e.  Top  ->  ( F : J -1-1-onto-> ( Clsd `  J
)  /\  `' F  =  ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) ) ) )
Distinct variable groups:    x, F    x, u, J    u, X, x
Allowed substitution hint:    F( u)

Proof of Theorem opncldf1
StepHypRef Expression
1 opncldf.2 . 2  |-  F  =  ( u  e.  J  |->  ( X  \  u
) )
2 opncldf.1 . . 3  |-  X  = 
U. J
32opncld 16786 . 2  |-  ( ( J  e.  Top  /\  u  e.  J )  ->  ( X  \  u
)  e.  ( Clsd `  J ) )
42cldopn 16784 . . 3  |-  ( x  e.  ( Clsd `  J
)  ->  ( X  \  x )  e.  J
)
54adantl 452 . 2  |-  ( ( J  e.  Top  /\  x  e.  ( Clsd `  J ) )  -> 
( X  \  x
)  e.  J )
62cldss 16782 . . . . . . 7  |-  ( x  e.  ( Clsd `  J
)  ->  x  C_  X
)
76ad2antll 709 . . . . . 6  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  x  C_  X
)
8 dfss4 3416 . . . . . 6  |-  ( x 
C_  X  <->  ( X  \  ( X  \  x
) )  =  x )
97, 8sylib 188 . . . . 5  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( X  \  ( X  \  x
) )  =  x )
109eqcomd 2301 . . . 4  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  x  =  ( X  \  ( X  \  x ) ) )
11 difeq2 3301 . . . . 5  |-  ( u  =  ( X  \  x )  ->  ( X  \  u )  =  ( X  \  ( X  \  x ) ) )
1211eqeq2d 2307 . . . 4  |-  ( u  =  ( X  \  x )  ->  (
x  =  ( X 
\  u )  <->  x  =  ( X  \  ( X  \  x ) ) ) )
1310, 12syl5ibrcom 213 . . 3  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( u  =  ( X  \  x )  ->  x  =  ( X  \  u ) ) )
142eltopss 16669 . . . . . . 7  |-  ( ( J  e.  Top  /\  u  e.  J )  ->  u  C_  X )
1514adantrr 697 . . . . . 6  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  u  C_  X
)
16 dfss4 3416 . . . . . 6  |-  ( u 
C_  X  <->  ( X  \  ( X  \  u
) )  =  u )
1715, 16sylib 188 . . . . 5  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( X  \  ( X  \  u
) )  =  u )
1817eqcomd 2301 . . . 4  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  u  =  ( X  \  ( X  \  u ) ) )
19 difeq2 3301 . . . . 5  |-  ( x  =  ( X  \  u )  ->  ( X  \  x )  =  ( X  \  ( X  \  u ) ) )
2019eqeq2d 2307 . . . 4  |-  ( x  =  ( X  \  u )  ->  (
u  =  ( X 
\  x )  <->  u  =  ( X  \  ( X  \  u ) ) ) )
2118, 20syl5ibrcom 213 . . 3  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( x  =  ( X  \  u )  ->  u  =  ( X  \  x ) ) )
2213, 21impbid 183 . 2  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( u  =  ( X  \  x )  <->  x  =  ( X  \  u
) ) )
231, 3, 5, 22f1ocnv2d 6084 1  |-  ( J  e.  Top  ->  ( F : J -1-1-onto-> ( Clsd `  J
)  /\  `' F  =  ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    \ cdif 3162    C_ wss 3165   U.cuni 3843    e. cmpt 4093   `'ccnv 4704   -1-1-onto->wf1o 5270   ` cfv 5271   Topctop 16647   Clsdccld 16769
This theorem is referenced by:  opncldf3  16839  cmpfi  17151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-cld 16772
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