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Theorem mrerintcl 14845
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 4322 . . . 4  |-  ( S  =  (/)  ->  ( X  i^i  |^| S )  =  X )
21adantl 466 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =  (/) )  ->  ( X  i^i  |^| S )  =  X )
3 mre1cl 14842 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
43ad2antrr 725 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =  (/) )  ->  X  e.  C )
52, 4eqeltrd 2555 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =  (/) )  ->  ( X  i^i  |^| S )  e.  C )
6 simp2 997 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  C_  C )
7 mresspw 14840 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  C  C_  ~P X )
873ad2ant1 1017 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  C  C_  ~P X )
96, 8sstrd 3514 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  C_  ~P X )
10 simp3 998 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  =/=  (/) )
11 rintn0 4416 . . . . 5  |-  ( ( S  C_  ~P X  /\  S  =/=  (/) )  -> 
( X  i^i  |^| S )  =  |^| S )
129, 10, 11syl2anc 661 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  ( X  i^i  |^| S )  = 
|^| S )
13 mreintcl 14843 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  |^| S  e.  C )
1412, 13eqeltrd 2555 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  ( X  i^i  |^| S )  e.  C )
15143expa 1196 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =/=  (/) )  ->  ( X  i^i  |^| S )  e.  C )
165, 15pm2.61dane 2785 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   |^|cint 4282   ` cfv 5586  Moorecmre 14830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-mre 14834
This theorem is referenced by:  mreacs  14906  topmtcl  29782
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