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Theorem mreriincl 15104
Description: The relative intersection of a family of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
mreriincl  |-  ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  ->  ( X  i^i  |^|_ y  e.  I  S )  e.  C
)
Distinct variable groups:    y, I    y, X    y, C
Allowed substitution hint:    S( y)

Proof of Theorem mreriincl
StepHypRef Expression
1 riin0 4344 . . . 4  |-  ( I  =  (/)  ->  ( X  i^i  |^|_ y  e.  I  S )  =  X )
21adantl 464 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  =  X )
3 mre1cl 15100 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
43ad2antrr 724 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =  (/) )  ->  X  e.  C )
52, 4eqeltrd 2490 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  e.  C
)
6 mress 15099 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  S  C_  X )
76ex 432 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  ( S  e.  C  ->  S  C_  X ) )
87ralimdv 2813 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  ( A. y  e.  I  S  e.  C  ->  A. y  e.  I  S  C_  X
) )
98imp 427 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  ->  A. y  e.  I  S  C_  X
)
10 riinn0 4345 . . . 4  |-  ( ( A. y  e.  I  S  C_  X  /\  I  =/=  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  =  |^|_ y  e.  I  S
)
119, 10sylan 469 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  =  |^|_ y  e.  I  S
)
12 simpll 752 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  C  e.  (Moore `  X )
)
13 simpr 459 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  I  =/=  (/) )
14 simplr 754 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  A. y  e.  I  S  e.  C )
15 mreiincl 15102 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^|_ y  e.  I  S  e.  C )
1612, 13, 14, 15syl3anc 1230 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  |^|_ y  e.  I  S  e.  C )
1711, 16eqeltrd 2490 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  e.  C
)
185, 17pm2.61dane 2721 1  |-  ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  ->  ( X  i^i  |^|_ y  e.  I  S )  e.  C
)
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    C_ wss 3413   (/)c0 3737   |^|_ciin 4271   ` cfv 5525  Moorecmre 15088
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
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-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-fv 5533  df-mre 15092
This theorem is referenced by:  acsfn1  15167  acsfn1c  15168  acsfn2  15169  acsfn1p  35493
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