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Theorem riincld 18790
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 4355 . . . 4  |-  ( A  =  (/)  ->  ( X  i^i  |^|_ x  e.  A  B )  =  X )
21adantl 466 . . 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 18781 . . . 4  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
54ad2antrr 725 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =  (/) )  ->  X  e.  ( Clsd `  J ) )
62, 5eqeltrd 2542 . 2  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =  (/) )  -> 
( X  i^i  |^|_ x  e.  A  B )  e.  ( Clsd `  J
) )
74ad2antrr 725 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  X  e.  ( Clsd `  J
) )
8 simpr 461 . . . 4  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  A  =/=  (/) )
9 simplr 754 . . . 4  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  A. x  e.  A  B  e.  ( Clsd `  J )
)
10 iincld 18785 . . . 4  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  |^|_ x  e.  A  B  e.  ( Clsd `  J )
)
118, 9, 10syl2anc 661 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  B  e.  ( Clsd `  J )
)
12 incld 18789 . . 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 661 . 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 2770 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799    i^i cin 3438   (/)c0 3748   U.cuni 4202   |^|_ciin 4283   ` cfv 5529   Topctop 18640   Clsdccld 18762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fn 5532  df-fv 5537  df-top 18645  df-cld 18765
This theorem is referenced by:  ptcld  19328  csscld  20903
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