MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riincld Structured version   Unicode version

Theorem riincld 19729
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 4344 . . . 4  |-  ( A  =  (/)  ->  ( X  i^i  |^|_ x  e.  A  B )  =  X )
21adantl 464 . . 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 19720 . . . 4  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
54ad2antrr 724 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =  (/) )  ->  X  e.  ( Clsd `  J ) )
62, 5eqeltrd 2490 . 2  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =  (/) )  -> 
( X  i^i  |^|_ x  e.  A  B )  e.  ( Clsd `  J
) )
74ad2antrr 724 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  X  e.  ( Clsd `  J
) )
8 simpr 459 . . . 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 19724 . . . 4  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  ( Clsd `  J
) )  ->  |^|_ x  e.  A  B  e.  ( Clsd `  J )
)
118, 9, 10syl2anc 659 . . 3  |-  ( ( ( J  e.  Top  /\ 
A. x  e.  A  B  e.  ( Clsd `  J ) )  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  B  e.  ( Clsd `  J )
)
12 incld 19728 . . 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 659 . 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 2721 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 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753    i^i cin 3412   (/)c0 3737   U.cuni 4190   |^|_ciin 4271   ` cfv 5525   Topctop 19578   Clsdccld 19701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-iota 5489  df-fun 5527  df-fn 5528  df-fv 5533  df-top 19583  df-cld 19704
This theorem is referenced by:  ptcld  20298  csscld  21873
  Copyright terms: Public domain W3C validator