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Theorem opncldf1 18529
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 18478 . 2  |-  ( ( J  e.  Top  /\  u  e.  J )  ->  ( X  \  u
)  e.  ( Clsd `  J ) )
42cldopn 18476 . . 3  |-  ( x  e.  ( Clsd `  J
)  ->  ( X  \  x )  e.  J
)
54adantl 463 . 2  |-  ( ( J  e.  Top  /\  x  e.  ( Clsd `  J ) )  -> 
( X  \  x
)  e.  J )
62cldss 18474 . . . . . . 7  |-  ( x  e.  ( Clsd `  J
)  ->  x  C_  X
)
76ad2antll 721 . . . . . 6  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  x  C_  X
)
8 dfss4 3572 . . . . . 6  |-  ( x 
C_  X  <->  ( X  \  ( X  \  x
) )  =  x )
97, 8sylib 196 . . . . 5  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( X  \  ( X  \  x
) )  =  x )
109eqcomd 2438 . . . 4  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  x  =  ( X  \  ( X  \  x ) ) )
11 difeq2 3456 . . . . 5  |-  ( u  =  ( X  \  x )  ->  ( X  \  u )  =  ( X  \  ( X  \  x ) ) )
1211eqeq2d 2444 . . . 4  |-  ( u  =  ( X  \  x )  ->  (
x  =  ( X 
\  u )  <->  x  =  ( X  \  ( X  \  x ) ) ) )
1310, 12syl5ibrcom 222 . . 3  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( u  =  ( X  \  x )  ->  x  =  ( X  \  u ) ) )
142eltopss 18361 . . . . . . 7  |-  ( ( J  e.  Top  /\  u  e.  J )  ->  u  C_  X )
1514adantrr 709 . . . . . 6  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  u  C_  X
)
16 dfss4 3572 . . . . . 6  |-  ( u 
C_  X  <->  ( X  \  ( X  \  u
) )  =  u )
1715, 16sylib 196 . . . . 5  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( X  \  ( X  \  u
) )  =  u )
1817eqcomd 2438 . . . 4  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  u  =  ( X  \  ( X  \  u ) ) )
19 difeq2 3456 . . . . 5  |-  ( x  =  ( X  \  u )  ->  ( X  \  x )  =  ( X  \  ( X  \  u ) ) )
2019eqeq2d 2444 . . . 4  |-  ( x  =  ( X  \  u )  ->  (
u  =  ( X 
\  x )  <->  u  =  ( X  \  ( X  \  u ) ) ) )
2118, 20syl5ibrcom 222 . . 3  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( x  =  ( X  \  u )  ->  u  =  ( X  \  x ) ) )
2213, 21impbid 191 . 2  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  ( u  =  ( X  \  x )  <->  x  =  ( X  \  u
) ) )
231, 3, 5, 22f1ocnv2d 6300 1  |-  ( J  e.  Top  ->  ( F : J -1-1-onto-> ( Clsd `  J
)  /\  `' F  =  ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755    \ cdif 3313    C_ wss 3316   U.cuni 4079    e. cmpt 4338   `'ccnv 4826   -1-1-onto->wf1o 5405   ` cfv 5406   Topctop 18339   Clsdccld 18461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-top 18344  df-cld 18464
This theorem is referenced by:  opncldf3  18531  cmpfi  18852
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