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Theorem opncldf2 17142
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 17090 . 2  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  ( X  \  A
)  e.  ( Clsd `  J ) )
4 difeq2 3452 . . 3  |-  ( u  =  A  ->  ( X  \  u )  =  ( X  \  A
) )
5 opncldf.2 . . 3  |-  F  =  ( u  e.  J  |->  ( X  \  u
) )
64, 5fvmptg 5797 . 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 1652    e. wcel 1725    \ cdif 3310   U.cuni 4008    e. cmpt 4259   ` cfv 5447   Topctop 16951   Clsdccld 17073
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fv 5455  df-top 16956  df-cld 17076
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