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Theorem mrerintcl 13499
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 3902 . . . 4  |-  ( S  =  (/)  ->  ( X  i^i  |^| S )  =  X )
21adantl 452 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =  (/) )  ->  ( X  i^i  |^| S )  =  X )
3 mre1cl 13496 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
43ad2antrr 706 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =  (/) )  ->  X  e.  C )
52, 4eqeltrd 2357 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =  (/) )  ->  ( X  i^i  |^| S )  e.  C )
6 simp2 956 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  C_  C )
7 mresspw 13494 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  C  C_  ~P X )
873ad2ant1 976 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  C  C_  ~P X )
96, 8sstrd 3189 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  C_  ~P X )
10 simp3 957 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  =/=  (/) )
11 rintn0 3992 . . . . 5  |-  ( ( S  C_  ~P X  /\  S  =/=  (/) )  -> 
( X  i^i  |^| S )  =  |^| S )
129, 10, 11syl2anc 642 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  ( X  i^i  |^| S )  = 
|^| S )
13 mreintcl 13497 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  |^| S  e.  C )
1412, 13eqeltrd 2357 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  ( X  i^i  |^| S )  e.  C )
15143expa 1151 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =/=  (/) )  ->  ( X  i^i  |^| S )  e.  C )
165, 15pm2.61dane 2524 1  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C )  ->  ( X  i^i  |^| S )  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   |^|cint 3862   ` cfv 5255  Moorecmre 13484
This theorem is referenced by:  mreacs  13560  topmtcl  26312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-mre 13488
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