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Theorem riincld 17110
Description: An indexed relative intersection of closed sets is closed. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
riincld  |-  ( ( J  e.  Top  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  ( X  i^i  |^|_ x  e.  A  B )  e.  (
Clsd `  J )
)
Distinct variable groups:    x, J    x, X    x, A
Allowed substitution hint:    B( x)

Proof of Theorem riincld
StepHypRef Expression
1 riin0 4166 . . . 4  |-  ( A  =  (/)  ->  ( X  i^i  |^|_ x  e.  A  B )  =  X )
21adantl 454 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =  (/) )  -> 
( X  i^i  |^|_ x  e.  A  B )  =  X )
3 clscld.1 . . . . 5  |-  X  = 
U. J
43topcld 17101 . . . 4  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
54ad2antrr 708 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =  (/) )  ->  X  e.  ( Clsd `  J ) )
62, 5eqeltrd 2512 . 2  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =  (/) )  -> 
( X  i^i  |^|_ x  e.  A  B )  e.  ( Clsd `  J
) )
74ad2antrr 708 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  X  e.  ( Clsd `  J
) )
8 simpr 449 . . . 4  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  A  =/=  (/) )
9 simplr 733 . . . 4  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  A. x  e.  A  B  e.  ( Clsd `  J )
)
10 iincld 17105 . . . 4  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  |^|_ x  e.  A  B  e.  ( Clsd `  J )
)
118, 9, 10syl2anc 644 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  B  e.  ( Clsd `  J )
)
12 incld 17109 . . 3  |-  ( ( X  e.  ( Clsd `  J )  /\  |^|_ x  e.  A  B  e.  ( Clsd `  J
) )  ->  ( X  i^i  |^|_ x  e.  A  B )  e.  (
Clsd `  J )
)
137, 11, 12syl2anc 644 . 2  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  ( X  i^i  |^|_ x  e.  A  B )  e.  (
Clsd `  J )
)
146, 13pm2.61dane 2684 1  |-  ( ( J  e.  Top  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  ( X  i^i  |^|_ x  e.  A  B )  e.  (
Clsd `  J )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707    i^i cin 3321   (/)c0 3630   U.cuni 4017   |^|_ciin 4096   ` cfv 5456   Topctop 16960   Clsdccld 17082
This theorem is referenced by:  ptcld  17647  csscld  19205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464  df-top 16965  df-cld 17085
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