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Theorem mrerintcl 15552
Description: The relative intersection of a set of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
mrerintcl  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C )  ->  ( X  i^i  |^| S )  e.  C )

Proof of Theorem mrerintcl
StepHypRef Expression
1 rint0 4289 . . . 4  |-  ( S  =  (/)  ->  ( X  i^i  |^| S )  =  X )
21adantl 472 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =  (/) )  ->  ( X  i^i  |^| S )  =  X )
3 mre1cl 15549 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
43ad2antrr 737 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =  (/) )  ->  X  e.  C )
52, 4eqeltrd 2540 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =  (/) )  ->  ( X  i^i  |^| S )  e.  C )
6 simp2 1015 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  C_  C )
7 mresspw 15547 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  C  C_  ~P X )
873ad2ant1 1035 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  C  C_  ~P X )
96, 8sstrd 3454 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  C_  ~P X )
10 simp3 1016 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  =/=  (/) )
11 rintn0 4386 . . . . 5  |-  ( ( S  C_  ~P X  /\  S  =/=  (/) )  -> 
( X  i^i  |^| S )  =  |^| S )
129, 10, 11syl2anc 671 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  ( X  i^i  |^| S )  = 
|^| S )
13 mreintcl 15550 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  |^| S  e.  C )
1412, 13eqeltrd 2540 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  ( X  i^i  |^| S )  e.  C )
15143expa 1215 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =/=  (/) )  ->  ( X  i^i  |^| S )  e.  C )
165, 15pm2.61dane 2723 1  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C )  ->  ( X  i^i  |^| S )  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633    i^i cin 3415    C_ wss 3416   (/)c0 3743   ~Pcpw 3963   |^|cint 4248   ` cfv 5601  Moorecmre 15537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-int 4249  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-mre 15541
This theorem is referenced by:  mreacs  15613  topmtcl  31068
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