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Theorem riincld 20051
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 4371 . . . 4  |-  ( A  =  (/)  ->  ( X  i^i  |^|_ x  e.  A  B )  =  X )
21adantl 468 . . 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 20042 . . . 4  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
54ad2antrr 731 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =  (/) )  ->  X  e.  ( Clsd `  J ) )
62, 5eqeltrd 2511 . 2  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =  (/) )  -> 
( X  i^i  |^|_ x  e.  A  B )  e.  ( Clsd `  J
) )
74ad2antrr 731 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  X  e.  ( Clsd `  J
) )
8 simpr 463 . . . 4  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  A  =/=  (/) )
9 simplr 761 . . . 4  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  A. x  e.  A  B  e.  ( Clsd `  J )
)
10 iincld 20046 . . . 4  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  |^|_ x  e.  A  B  e.  ( Clsd `  J )
)
118, 9, 10syl2anc 666 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  B  e.  ( Clsd `  J )
)
12 incld 20050 . . 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 666 . 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 2743 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 371    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776    i^i cin 3436   (/)c0 3762   U.cuni 4217   |^|_ciin 4298   ` cfv 5599   Topctop 19909   Clsdccld 20023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-iota 5563  df-fun 5601  df-fn 5602  df-fv 5607  df-top 19913  df-cld 20026
This theorem is referenced by:  ptcld  20620  csscld  22212
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