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Theorem mrerintcl 15013
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 4329 . . . 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 15010 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
43ad2antrr 725 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  S  C_  C )  /\  S  =  (/) )  ->  X  e.  C )
52, 4eqeltrd 2545 . 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 15008 . . . . . . 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 3509 . . . . 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 4426 . . . . 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 15011 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  |^| S  e.  C )
1412, 13eqeltrd 2545 . . 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 2775 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 1395    e. wcel 1819    =/= wne 2652    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   |^|cint 4288   ` cfv 5594  Moorecmre 14998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-mre 15002
This theorem is referenced by:  mreacs  15074  topmtcl  30343
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