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Theorem iuncld 17025
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 3514 . . . 4  |-  ( X 
\  ( X  i^i  |^|_
x  e.  A  ( X  \  B ) ) )  =  ( X  \  |^|_ x  e.  A  ( X  \  B ) )
2 iundif2 4092 . . . 4  |-  U_ x  e.  A  ( X  \  ( X  \  B
) )  =  ( X  \  |^|_ x  e.  A  ( X  \  B ) )
31, 2eqtr4i 2403 . . 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 17009 . . . . . . 7  |-  ( B  e.  ( Clsd `  J
)  ->  B  C_  X
)
6 dfss4 3511 . . . . . . 7  |-  ( B 
C_  X  <->  ( X  \  ( X  \  B
) )  =  B )
75, 6sylib 189 . . . . . 6  |-  ( B  e.  ( Clsd `  J
)  ->  ( X  \  ( X  \  B
) )  =  B )
87ralimi 2717 . . . . 5  |-  ( A. x  e.  A  B  e.  ( Clsd `  J
)  ->  A. x  e.  A  ( X  \  ( X  \  B
) )  =  B )
983ad2ant3 980 . . . 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 4044 . . . 4  |-  ( A. x  e.  A  ( X  \  ( X  \  B ) )  =  B  ->  U_ x  e.  A  ( X  \ 
( X  \  B
) )  =  U_ x  e.  A  B
)
119, 10syl 16 . . 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 2424 . 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 957 . . 3  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  J  e.  Top )
144cldopn 17011 . . . . 5  |-  ( B  e.  ( Clsd `  J
)  ->  ( X  \  B )  e.  J
)
1514ralimi 2717 . . . 4  |-  ( A. x  e.  A  B  e.  ( Clsd `  J
)  ->  A. x  e.  A  ( X  \  B )  e.  J
)
164riinopn 16897 . . . 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 1219 . . 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 17013 . . 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 643 . 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 2455 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 936    = wceq 1649    e. wcel 1717   A.wral 2642    \ cdif 3253    i^i cin 3255    C_ wss 3256   U.cuni 3950   U_ciun 4028   |^|_ciin 4029   ` cfv 5387   Fincfn 7038   Topctop 16874   Clsdccld 16996
This theorem is referenced by:  unicld  17026  t1ficld  17306
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-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-en 7039  df-dom 7040  df-fin 7042  df-top 16879  df-cld 16999
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