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Theorem opncldf2 17072
Description: The values 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
opncldf2  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  ( F `  A
)  =  ( X 
\  A ) )
Distinct variable groups:    u, A    u, J    u, X
Allowed substitution hint:    F( u)

Proof of Theorem opncldf2
StepHypRef Expression
1 simpr 448 . 2  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  A  e.  J )
2 opncldf.1 . . 3  |-  X  = 
U. J
32opncld 17020 . 2  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  ( X  \  A
)  e.  ( Clsd `  J ) )
4 difeq2 3402 . . 3  |-  ( u  =  A  ->  ( X  \  u )  =  ( X  \  A
) )
5 opncldf.2 . . 3  |-  F  =  ( u  e.  J  |->  ( X  \  u
) )
64, 5fvmptg 5743 . 2  |-  ( ( A  e.  J  /\  ( X  \  A )  e.  ( Clsd `  J
) )  ->  ( F `  A )  =  ( X  \  A ) )
71, 3, 6syl2anc 643 1  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  ( F `  A
)  =  ( X 
\  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    \ cdif 3260   U.cuni 3957    e. cmpt 4207   ` 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-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
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-iota 5358  df-fun 5396  df-fv 5402  df-top 16886  df-cld 17006
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