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Theorem opncldf1 18710
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 18659 . 2  |-  ( ( J  e.  Top  /\  u  e.  J )  ->  ( X  \  u
)  e.  ( Clsd `  J ) )
42cldopn 18657 . . 3  |-  ( x  e.  ( Clsd `  J
)  ->  ( X  \  x )  e.  J
)
54adantl 466 . 2  |-  ( ( J  e.  Top  /\  x  e.  ( Clsd `  J ) )  -> 
( X  \  x
)  e.  J )
62cldss 18655 . . . . . . 7  |-  ( x  e.  ( Clsd `  J
)  ->  x  C_  X
)
76ad2antll 728 . . . . . 6  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  x  C_  X
)
8 dfss4 3605 . . . . . 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 2448 . . . 4  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  x  =  ( X  \  ( X  \  x ) ) )
11 difeq2 3489 . . . . 5  |-  ( u  =  ( X  \  x )  ->  ( X  \  u )  =  ( X  \  ( X  \  x ) ) )
1211eqeq2d 2454 . . . 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 18542 . . . . . . 7  |-  ( ( J  e.  Top  /\  u  e.  J )  ->  u  C_  X )
1514adantrr 716 . . . . . 6  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  u  C_  X
)
16 dfss4 3605 . . . . . 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 2448 . . . 4  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  u  =  ( X  \  ( X  \  u ) ) )
19 difeq2 3489 . . . . 5  |-  ( x  =  ( X  \  u )  ->  ( X  \  x )  =  ( X  \  ( X  \  u ) ) )
2019eqeq2d 2454 . . . 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 6332 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 1369    e. wcel 1756    \ cdif 3346    C_ wss 3349   U.cuni 4112    e. cmpt 4371   `'ccnv 4860   -1-1-onto->wf1o 5438   ` cfv 5439   Topctop 18520   Clsdccld 18642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-top 18525  df-cld 18645
This theorem is referenced by:  opncldf3  18712  cmpfi  19033
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