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Theorem opncldf3 17073
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 17013 . . . 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 17071 . . . . 5  |-  ( J  e.  Top  ->  ( F : J -1-1-onto-> ( Clsd `  J
)  /\  `' F  =  ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) ) ) )
54simprd 450 . . . 4  |-  ( J  e.  Top  ->  `' F  =  ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) ) )
61, 5syl 16 . . 3  |-  ( B  e.  ( Clsd `  J
)  ->  `' F  =  ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) ) )
76fveq1d 5670 . 2  |-  ( B  e.  ( Clsd `  J
)  ->  ( `' F `  B )  =  ( ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) ) `  B ) )
82cldopn 17018 . . 3  |-  ( B  e.  ( Clsd `  J
)  ->  ( X  \  B )  e.  J
)
9 difeq2 3402 . . . 4  |-  ( x  =  B  ->  ( X  \  x )  =  ( X  \  B
) )
10 eqid 2387 . . . 4  |-  ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) )  =  ( x  e.  (
Clsd `  J )  |->  ( X  \  x
) )
119, 10fvmptg 5743 . . 3  |-  ( ( B  e.  ( Clsd `  J )  /\  ( X  \  B )  e.  J )  ->  (
( x  e.  (
Clsd `  J )  |->  ( X  \  x
) ) `  B
)  =  ( X 
\  B ) )
128, 11mpdan 650 . 2  |-  ( B  e.  ( Clsd `  J
)  ->  ( (
x  e.  ( Clsd `  J )  |->  ( X 
\  x ) ) `
 B )  =  ( X  \  B
) )
137, 12eqtrd 2419 1  |-  ( B  e.  ( Clsd `  J
)  ->  ( `' F `  B )  =  ( X  \  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    \ cdif 3260   U.cuni 3957    e. cmpt 4207   `'ccnv 4817   -1-1-onto->wf1o 5393   ` cfv 5394   Topctop 16881   Clsdccld 17003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-top 16886  df-cld 17006
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