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Theorem opncldf3 17140
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 17080 . . . 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 17138 . . . . 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 5722 . 2  |-  ( B  e.  ( Clsd `  J
)  ->  ( `' F `  B )  =  ( ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) ) `  B ) )
82cldopn 17085 . . 3  |-  ( B  e.  ( Clsd `  J
)  ->  ( X  \  B )  e.  J
)
9 difeq2 3451 . . . 4  |-  ( x  =  B  ->  ( X  \  x )  =  ( X  \  B
) )
10 eqid 2435 . . . 4  |-  ( x  e.  ( Clsd `  J
)  |->  ( X  \  x ) )  =  ( x  e.  (
Clsd `  J )  |->  ( X  \  x
) )
119, 10fvmptg 5796 . . 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 2467 1  |-  ( B  e.  ( Clsd `  J
)  ->  ( `' F `  B )  =  ( X  \  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    \ cdif 3309   U.cuni 4007    e. cmpt 4258   `'ccnv 4869   -1-1-onto->wf1o 5445   ` cfv 5446   Topctop 16948   Clsdccld 17070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-top 16953  df-cld 17073
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