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Theorem opncldf2 16838
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 447 . 2  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  A  e.  J )
2 opncldf.1 . . 3  |-  X  = 
U. J
32opncld 16786 . 2  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  ( X  \  A
)  e.  ( Clsd `  J ) )
4 difeq2 3301 . . 3  |-  ( u  =  A  ->  ( X  \  u )  =  ( X  \  A
) )
5 opncldf.2 . . 3  |-  F  =  ( u  e.  J  |->  ( X  \  u
) )
64, 5fvmptg 5616 . 2  |-  ( ( A  e.  J  /\  ( X  \  A )  e.  ( Clsd `  J
) )  ->  ( F `  A )  =  ( X  \  A ) )
71, 3, 6syl2anc 642 1  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  ( F `  A
)  =  ( X 
\  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    \ cdif 3162   U.cuni 3843    e. cmpt 4093   ` 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-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
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-iota 5235  df-fun 5273  df-fv 5279  df-top 16652  df-cld 16772
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