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Theorem mreriincl 13710
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 4077 . . . 4  |-  ( I  =  (/)  ->  ( X  i^i  |^|_ y  e.  I  S )  =  X )
21adantl 452 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  =  X )
3 mre1cl 13706 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
43ad2antrr 706 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =  (/) )  ->  X  e.  C )
52, 4eqeltrd 2440 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  e.  C
)
6 mress 13705 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  S  C_  X )
76ex 423 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  ( S  e.  C  ->  S  C_  X ) )
87ralimdv 2707 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  ( A. y  e.  I  S  e.  C  ->  A. y  e.  I  S  C_  X
) )
98imp 418 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  ->  A. y  e.  I  S  C_  X
)
10 riinn0 4078 . . . 4  |-  ( ( A. y  e.  I  S  C_  X  /\  I  =/=  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  =  |^|_ y  e.  I  S
)
119, 10sylan 457 . . 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 730 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  C  e.  (Moore `  X )
)
13 simpr 447 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  I  =/=  (/) )
14 simplr 731 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  A. y  e.  I  S  e.  C )
15 mreiincl 13708 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^|_ y  e.  I  S  e.  C )
1612, 13, 14, 15syl3anc 1183 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  |^|_ y  e.  I  S  e.  C )
1711, 16eqeltrd 2440 . 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 2607 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 358    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628    i^i cin 3237    C_ wss 3238   (/)c0 3543   |^|_ciin 4008   ` cfv 5358  Moorecmre 13694
This theorem is referenced by:  acsfn1  13773  acsfn1c  13774  acsfn2  13775  acsfn1p  27013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-int 3965  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-mre 13698
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