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Theorem submre 13835
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 3564 . . 3  |-  ( C  i^i  ~P A ) 
C_  ~P A
21a1i 11 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  ( C  i^i  ~P A ) 
C_  ~P A )
3 simpr 449 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  A  e.  C )
4 pwidg 3813 . . . 4  |-  ( A  e.  C  ->  A  e.  ~P A )
54adantl 454 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  A  e.  ~P A )
6 elin 3532 . . 3  |-  ( A  e.  ( C  i^i  ~P A )  <->  ( A  e.  C  /\  A  e. 
~P A ) )
73, 5, 6sylanbrc 647 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  A  e.  ( C  i^i  ~P A ) )
8 simp1l 982 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  C  e.  (Moore `  X )
)
9 inss1 3563 . . . . . 6  |-  ( C  i^i  ~P A ) 
C_  C
10 sstr 3358 . . . . . 6  |-  ( ( x  C_  ( C  i^i  ~P A )  /\  ( C  i^i  ~P A
)  C_  C )  ->  x  C_  C )
119, 10mpan2 654 . . . . 5  |-  ( x 
C_  ( C  i^i  ~P A )  ->  x  C_  C )
12113ad2ant2 980 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  C_  C )
13 simp3 960 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  =/=  (/) )
14 mreintcl 13825 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  e.  C )
158, 12, 13, 14syl3anc 1185 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  e.  C )
16 sstr 3358 . . . . . . . 8  |-  ( ( x  C_  ( C  i^i  ~P A )  /\  ( C  i^i  ~P A
)  C_  ~P A
)  ->  x  C_  ~P A )
171, 16mpan2 654 . . . . . . 7  |-  ( x 
C_  ( C  i^i  ~P A )  ->  x  C_ 
~P A )
18173ad2ant2 980 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  C_ 
~P A )
19 intssuni2 4077 . . . . . 6  |-  ( ( x  C_  ~P A  /\  x  =/=  (/) )  ->  |^| x  C_  U. ~P A )
2018, 13, 19syl2anc 644 . . . . 5  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  C_ 
U. ~P A )
21 unipw 4417 . . . . 5  |-  U. ~P A  =  A
2220, 21syl6sseq 3396 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  C_  A )
23 elpw2g 4366 . . . . . 6  |-  ( A  e.  C  ->  ( |^| x  e.  ~P A 
<-> 
|^| x  C_  A
) )
2423adantl 454 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  ( |^| x  e.  ~P A 
<-> 
|^| x  C_  A
) )
25243ad2ant1 979 . . . 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 225 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  e.  ~P A )
27 elin 3532 . . 3  |-  ( |^| x  e.  ( C  i^i  ~P A )  <->  ( |^| x  e.  C  /\  |^| x  e.  ~P A
) )
2815, 26, 27sylanbrc 647 . 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 13832 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 178    /\ wa 360    /\ w3a 937    e. wcel 1726    =/= wne 2601    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   U.cuni 4017   |^|cint 4052   ` cfv 5457  Moorecmre 13812
This theorem is referenced by:  submrc  13858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
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 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-mre 13816
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