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Theorem riincld 16781
Description: A 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 3975 . . . 4  |-  ( A  =  (/)  ->  ( X  i^i  |^|_ x  e.  A  B )  =  X )
21adantl 452 . . 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 16772 . . . 4  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
54ad2antrr 706 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =  (/) )  ->  X  e.  ( Clsd `  J ) )
62, 5eqeltrd 2357 . 2  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =  (/) )  -> 
( X  i^i  |^|_ x  e.  A  B )  e.  ( Clsd `  J
) )
74ad2antrr 706 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  X  e.  ( Clsd `  J
) )
8 simpr 447 . . . 4  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  A  =/=  (/) )
9 simplr 731 . . . 4  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  A. x  e.  A  B  e.  ( Clsd `  J )
)
10 iincld 16776 . . . 4  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  |^|_ x  e.  A  B  e.  ( Clsd `  J )
)
118, 9, 10syl2anc 642 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  B  e.  ( Clsd `  J )
)
12 incld 16780 . . 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 642 . 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 2524 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 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    i^i cin 3151   (/)c0 3455   U.cuni 3827   |^|_ciin 3906   ` cfv 5255   Topctop 16631   Clsdccld 16753
This theorem is referenced by:  ptcld  17307  csscld  18676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-top 16636  df-cld 16756
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