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Theorem mrerintcl 13824
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 4092 . . . 4  |-  ( S  =  (/)  ->  ( X  i^i  |^| S )  =  X )
21adantl 454 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =  (/) )  ->  ( X  i^i  |^| S )  =  X )
3 mre1cl 13821 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
43ad2antrr 708 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =  (/) )  ->  X  e.  C )
52, 4eqeltrd 2512 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =  (/) )  ->  ( X  i^i  |^| S )  e.  C )
6 simp2 959 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  C_  C )
7 mresspw 13819 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  C  C_  ~P X )
873ad2ant1 979 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  C  C_  ~P X )
96, 8sstrd 3360 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  C_  ~P X )
10 simp3 960 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  =/=  (/) )
11 rintn0 4183 . . . . 5  |-  ( ( S  C_  ~P X  /\  S  =/=  (/) )  -> 
( X  i^i  |^| S )  =  |^| S )
129, 10, 11syl2anc 644 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  ( X  i^i  |^| S )  = 
|^| S )
13 mreintcl 13822 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  |^| S  e.  C )
1412, 13eqeltrd 2512 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  ( X  i^i  |^| S )  e.  C )
15143expa 1154 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =/=  (/) )  ->  ( X  i^i  |^| S )  e.  C )
165, 15pm2.61dane 2684 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   |^|cint 4052   ` cfv 5456  Moorecmre 13809
This theorem is referenced by:  mreacs  13885  topmtcl  26394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
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-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-mre 13813
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