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Theorem opncldf3 16839
Description: The values of the converse/inverse of the open-closed bijection. (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
opncldf3  |-  ( B  e.  ( Clsd `  J
)  ->  ( `' F `  B )  =  ( X  \  B ) )
Distinct variable groups:    u, J    u, X
Allowed substitution hints:    B( u)    F( u)

Proof of Theorem opncldf3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cldrcl 16779 . . . 4  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
2 opncldf.1 . . . . . 6  |-  X  = 
U. J
3 opncldf.2 . . . . . 6  |-  F  =  ( u  e.  J  |->  ( X  \  u
) )
42, 3opncldf1 16837 . . . . 5  |-  ( J  e.  Top  ->  ( F : J -1-1-onto-> ( Clsd `  J
)  /\  `' F  =  ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) ) ) )
54simprd 449 . . . 4  |-  ( J  e.  Top  ->  `' F  =  ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) ) )
61, 5syl 15 . . 3  |-  ( B  e.  ( Clsd `  J
)  ->  `' F  =  ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) ) )
76fveq1d 5543 . 2  |-  ( B  e.  ( Clsd `  J
)  ->  ( `' F `  B )  =  ( ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) ) `  B ) )
82cldopn 16784 . . 3  |-  ( B  e.  ( Clsd `  J
)  ->  ( X  \  B )  e.  J
)
9 difeq2 3301 . . . 4  |-  ( x  =  B  ->  ( X  \  x )  =  ( X  \  B
) )
10 eqid 2296 . . . 4  |-  ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) )  =  ( x  e.  (
Clsd `  J )  |->  ( X  \  x
) )
119, 10fvmptg 5616 . . 3  |-  ( ( B  e.  ( Clsd `  J )  /\  ( X  \  B )  e.  J )  ->  (
( x  e.  (
Clsd `  J )  |->  ( X  \  x
) ) `  B
)  =  ( X 
\  B ) )
128, 11mpdan 649 . 2  |-  ( B  e.  ( Clsd `  J
)  ->  ( (
x  e.  ( Clsd `  J )  |->  ( X 
\  x ) ) `
 B )  =  ( X  \  B
) )
137, 12eqtrd 2328 1  |-  ( B  e.  ( Clsd `  J
)  ->  ( `' F `  B )  =  ( X  \  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    \ cdif 3162   U.cuni 3843    e. cmpt 4093   `'ccnv 4704   -1-1-onto->wf1o 5270   ` cfv 5271   Topctop 16647   Clsdccld 16769
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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