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Theorem opncldf1 16821
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 16770 . 2  |-  ( ( J  e.  Top  /\  u  e.  J )  ->  ( X  \  u
)  e.  ( Clsd `  J ) )
42cldopn 16768 . . 3  |-  ( x  e.  ( Clsd `  J
)  ->  ( X  \  x )  e.  J
)
54adantl 452 . 2  |-  ( ( J  e.  Top  /\  x  e.  ( Clsd `  J ) )  -> 
( X  \  x
)  e.  J )
62cldss 16766 . . . . . . 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 3403 . . . . . 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 2288 . . . 4  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  x  =  ( X  \  ( X  \  x ) ) )
11 difeq2 3288 . . . . 5  |-  ( u  =  ( X  \  x )  ->  ( X  \  u )  =  ( X  \  ( X  \  x ) ) )
1211eqeq2d 2294 . . . 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 16653 . . . . . . 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 3403 . . . . . 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 2288 . . . 4  |-  ( ( J  e.  Top  /\  ( u  e.  J  /\  x  e.  ( Clsd `  J ) ) )  ->  u  =  ( X  \  ( X  \  u ) ) )
19 difeq2 3288 . . . . 5  |-  ( x  =  ( X  \  u )  ->  ( X  \  x )  =  ( X  \  ( X  \  u ) ) )
2019eqeq2d 2294 . . . 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 6068 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 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   U.cuni 3827    e. cmpt 4077   `'ccnv 4688   -1-1-onto->wf1o 5254   ` cfv 5255   Topctop 16631   Clsdccld 16753
This theorem is referenced by:  opncldf3  16823  cmpfi  17135
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-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-rab 2552  df-v 2790  df-sbc 2992  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-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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-cld 16756
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