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Theorem opncldf1 17150
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 17099 . 2  |-  ( ( J  e.  Top  /\  u  e.  J )  ->  ( X  \  u
)  e.  ( Clsd `  J ) )
42cldopn 17097 . . 3  |-  ( x  e.  ( Clsd `  J
)  ->  ( X  \  x )  e.  J
)
54adantl 454 . 2  |-  ( ( J  e.  Top  /\  x  e.  ( Clsd `  J ) )  -> 
( X  \  x
)  e.  J )
62cldss 17095 . . . . . . 7  |-  ( x  e.  ( Clsd `  J
)  ->  x  C_  X
)
76ad2antll 711 . . . . . 6  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  x  C_  X
)
8 dfss4 3577 . . . . . 6  |-  ( x 
C_  X  <->  ( X  \  ( X  \  x
) )  =  x )
97, 8sylib 190 . . . . 5  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( X  \  ( X  \  x
) )  =  x )
109eqcomd 2443 . . . 4  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  x  =  ( X  \  ( X  \  x ) ) )
11 difeq2 3461 . . . . 5  |-  ( u  =  ( X  \  x )  ->  ( X  \  u )  =  ( X  \  ( X  \  x ) ) )
1211eqeq2d 2449 . . . 4  |-  ( u  =  ( X  \  x )  ->  (
x  =  ( X 
\  u )  <->  x  =  ( X  \  ( X  \  x ) ) ) )
1310, 12syl5ibrcom 215 . . 3  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( u  =  ( X  \  x )  ->  x  =  ( X  \  u ) ) )
142eltopss 16982 . . . . . . 7  |-  ( ( J  e.  Top  /\  u  e.  J )  ->  u  C_  X )
1514adantrr 699 . . . . . 6  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  u  C_  X
)
16 dfss4 3577 . . . . . 6  |-  ( u 
C_  X  <->  ( X  \  ( X  \  u
) )  =  u )
1715, 16sylib 190 . . . . 5  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( X  \  ( X  \  u
) )  =  u )
1817eqcomd 2443 . . . 4  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  u  =  ( X  \  ( X  \  u ) ) )
19 difeq2 3461 . . . . 5  |-  ( x  =  ( X  \  u )  ->  ( X  \  x )  =  ( X  \  ( X  \  u ) ) )
2019eqeq2d 2449 . . . 4  |-  ( x  =  ( X  \  u )  ->  (
u  =  ( X 
\  x )  <->  u  =  ( X  \  ( X  \  u ) ) ) )
2118, 20syl5ibrcom 215 . . 3  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( x  =  ( X  \  u )  ->  u  =  ( X  \  x ) ) )
2213, 21impbid 185 . 2  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( u  =  ( X  \  x )  <->  x  =  ( X  \  u
) ) )
231, 3, 5, 22f1ocnv2d 6297 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 360    = wceq 1653    e. wcel 1726    \ cdif 3319    C_ wss 3322   U.cuni 4017    e. cmpt 4268   `'ccnv 4879   -1-1-onto->wf1o 5455   ` cfv 5456   Topctop 16960   Clsdccld 17082
This theorem is referenced by:  opncldf3  17152  cmpfi  17473
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-top 16965  df-cld 17085
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