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Theorem iuncld 16798
Description: A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
iuncld  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  U_ x  e.  A  B  e.  ( Clsd `  J )
)
Distinct variable groups:    x, J    x, X    x, A
Allowed substitution hint:    B( x)

Proof of Theorem iuncld
StepHypRef Expression
1 difin 3419 . . . 4  |-  ( X 
\  ( X  i^i  |^|_
x  e.  A  ( X  \  B ) ) )  =  ( X  \  |^|_ x  e.  A  ( X  \  B ) )
2 iundif2 3985 . . . 4  |-  U_ x  e.  A  ( X  \  ( X  \  B
) )  =  ( X  \  |^|_ x  e.  A  ( X  \  B ) )
31, 2eqtr4i 2319 . . 3  |-  ( X 
\  ( X  i^i  |^|_
x  e.  A  ( X  \  B ) ) )  =  U_ x  e.  A  ( X  \  ( X  \  B ) )
4 clscld.1 . . . . . . . 8  |-  X  = 
U. J
54cldss 16782 . . . . . . 7  |-  ( B  e.  ( Clsd `  J
)  ->  B  C_  X
)
6 dfss4 3416 . . . . . . 7  |-  ( B 
C_  X  <->  ( X  \  ( X  \  B
) )  =  B )
75, 6sylib 188 . . . . . 6  |-  ( B  e.  ( Clsd `  J
)  ->  ( X  \  ( X  \  B
) )  =  B )
87ralimi 2631 . . . . 5  |-  ( A. x  e.  A  B  e.  ( Clsd `  J
)  ->  A. x  e.  A  ( X  \  ( X  \  B
) )  =  B )
983ad2ant3 978 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  A. x  e.  A  ( X  \  ( X  \  B
) )  =  B )
10 iuneq2 3937 . . . 4  |-  ( A. x  e.  A  ( X  \  ( X  \  B ) )  =  B  ->  U_ x  e.  A  ( X  \ 
( X  \  B
) )  =  U_ x  e.  A  B
)
119, 10syl 15 . . 3  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  U_ x  e.  A  ( X  \  ( X  \  B
) )  =  U_ x  e.  A  B
)
123, 11syl5eq 2340 . 2  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  ( X  \  ( X  i^i  |^|_
x  e.  A  ( X  \  B ) ) )  =  U_ x  e.  A  B
)
13 simp1 955 . . 3  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  J  e.  Top )
144cldopn 16784 . . . . 5  |-  ( B  e.  ( Clsd `  J
)  ->  ( X  \  B )  e.  J
)
1514ralimi 2631 . . . 4  |-  ( A. x  e.  A  B  e.  ( Clsd `  J
)  ->  A. x  e.  A  ( X  \  B )  e.  J
)
164riinopn 16670 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  ( X  \  B )  e.  J )  ->  ( X  i^i  |^|_ x  e.  A  ( X  \  B ) )  e.  J )
1715, 16syl3an3 1217 . . 3  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  ( X  i^i  |^|_ x  e.  A  ( X  \  B ) )  e.  J )
184opncld 16786 . . 3  |-  ( ( J  e.  Top  /\  ( X  i^i  |^|_ x  e.  A  ( X  \  B ) )  e.  J )  ->  ( X  \  ( X  i^i  |^|_
x  e.  A  ( X  \  B ) ) )  e.  (
Clsd `  J )
)
1913, 17, 18syl2anc 642 . 2  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  ( X  \  ( X  i^i  |^|_
x  e.  A  ( X  \  B ) ) )  e.  (
Clsd `  J )
)
2012, 19eqeltrrd 2371 1  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  U_ x  e.  A  B  e.  ( Clsd `  J )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    \ cdif 3162    i^i cin 3164    C_ wss 3165   U.cuni 3843   U_ciun 3921   |^|_ciin 3922   ` cfv 5271   Fincfn 6879   Topctop 16647   Clsdccld 16769
This theorem is referenced by:  unicld  16799  t1ficld  17071
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-13 1698  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  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-fin 6883  df-top 16652  df-cld 16772
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