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Theorem mreriincl 13828
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 4167 . . . 4  |-  ( I  =  (/)  ->  ( X  i^i  |^|_ y  e.  I  S )  =  X )
21adantl 454 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  =  X )
3 mre1cl 13824 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
43ad2antrr 708 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =  (/) )  ->  X  e.  C )
52, 4eqeltrd 2512 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  e.  C
)
6 mress 13823 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  S  C_  X )
76ex 425 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  ( S  e.  C  ->  S  C_  X ) )
87ralimdv 2787 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  ( A. y  e.  I  S  e.  C  ->  A. y  e.  I  S  C_  X
) )
98imp 420 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  ->  A. y  e.  I  S  C_  X
)
10 riinn0 4168 . . . 4  |-  ( ( A. y  e.  I  S  C_  X  /\  I  =/=  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  =  |^|_ y  e.  I  S
)
119, 10sylan 459 . . 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 732 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  C  e.  (Moore `  X )
)
13 simpr 449 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  I  =/=  (/) )
14 simplr 733 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  A. y  e.  I  S  e.  C )
15 mreiincl 13826 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^|_ y  e.  I  S  e.  C )
1612, 13, 14, 15syl3anc 1185 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  |^|_ y  e.  I  S  e.  C )
1711, 16eqeltrd 2512 . 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 2684 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707    i^i cin 3321    C_ wss 3322   (/)c0 3630   |^|_ciin 4096   ` cfv 5457  Moorecmre 13812
This theorem is referenced by:  acsfn1  13891  acsfn1c  13892  acsfn2  13893  acsfn1p  27498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-mre 13816
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