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Theorem submre 13717
Description: The subcollection of a closed set system below a given closed set is itself a closed set system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Assertion
Ref Expression
submre  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  ( C  i^i  ~P A )  e.  (Moore `  A
) )

Proof of Theorem submre
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 inss2 3478 . . 3  |-  ( C  i^i  ~P A ) 
C_  ~P A
21a1i 10 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  ( C  i^i  ~P A ) 
C_  ~P A )
3 simpr 447 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  A  e.  C )
4 pwidg 3726 . . . 4  |-  ( A  e.  C  ->  A  e.  ~P A )
54adantl 452 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  A  e.  ~P A )
6 elin 3446 . . 3  |-  ( A  e.  ( C  i^i  ~P A )  <->  ( A  e.  C  /\  A  e. 
~P A ) )
73, 5, 6sylanbrc 645 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  A  e.  ( C  i^i  ~P A ) )
8 simp1l 980 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  C  e.  (Moore `  X )
)
9 inss1 3477 . . . . . 6  |-  ( C  i^i  ~P A ) 
C_  C
10 sstr 3273 . . . . . 6  |-  ( ( x  C_  ( C  i^i  ~P A )  /\  ( C  i^i  ~P A
)  C_  C )  ->  x  C_  C )
119, 10mpan2 652 . . . . 5  |-  ( x 
C_  ( C  i^i  ~P A )  ->  x  C_  C )
12113ad2ant2 978 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  C_  C )
13 simp3 958 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  =/=  (/) )
14 mreintcl 13707 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  e.  C )
158, 12, 13, 14syl3anc 1183 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  e.  C )
16 sstr 3273 . . . . . . . 8  |-  ( ( x  C_  ( C  i^i  ~P A )  /\  ( C  i^i  ~P A
)  C_  ~P A
)  ->  x  C_  ~P A )
171, 16mpan2 652 . . . . . . 7  |-  ( x 
C_  ( C  i^i  ~P A )  ->  x  C_ 
~P A )
18173ad2ant2 978 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  C_ 
~P A )
19 intssuni2 3989 . . . . . 6  |-  ( ( x  C_  ~P A  /\  x  =/=  (/) )  ->  |^| x  C_  U. ~P A )
2018, 13, 19syl2anc 642 . . . . 5  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  C_ 
U. ~P A )
21 unipw 4327 . . . . 5  |-  U. ~P A  =  A
2220, 21syl6sseq 3310 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  C_  A )
23 elpw2g 4276 . . . . . 6  |-  ( A  e.  C  ->  ( |^| x  e.  ~P A 
<-> 
|^| x  C_  A
) )
2423adantl 452 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  ( |^| x  e.  ~P A 
<-> 
|^| x  C_  A
) )
25243ad2ant1 977 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  ( |^| x  e.  ~P A 
<-> 
|^| x  C_  A
) )
2622, 25mpbird 223 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  e.  ~P A )
27 elin 3446 . . 3  |-  ( |^| x  e.  ( C  i^i  ~P A )  <->  ( |^| x  e.  C  /\  |^| x  e.  ~P A
) )
2815, 26, 27sylanbrc 645 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  e.  ( C  i^i  ~P A ) )
292, 7, 28ismred 13714 1  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  ( C  i^i  ~P A )  e.  (Moore `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    e. wcel 1715    =/= wne 2529    i^i cin 3237    C_ wss 3238   (/)c0 3543   ~Pcpw 3714   U.cuni 3929   |^|cint 3964   ` cfv 5358  Moorecmre 13694
This theorem is referenced by:  submrc  13740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-int 3965  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-mre 13698
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