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Theorem submre 13507
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 3390 . . 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 3637 . . . 4  |-  ( A  e.  C  ->  A  e.  ~P A )
54adantl 452 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  A  e.  ~P A )
6 elin 3358 . . 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 979 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  C  e.  (Moore `  X )
)
9 inss1 3389 . . . . . 6  |-  ( C  i^i  ~P A ) 
C_  C
10 sstr 3187 . . . . . 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 977 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  C_  C )
13 simp3 957 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  =/=  (/) )
14 mreintcl 13497 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  e.  C )
158, 12, 13, 14syl3anc 1182 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  e.  C )
16 sstr 3187 . . . . . . . 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 977 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  C_ 
~P A )
19 intssuni2 3887 . . . . . 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 4224 . . . . 5  |-  U. ~P A  =  A
2220, 21syl6sseq 3224 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  C_  A )
23 elpw2g 4174 . . . . . 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 976 . . . 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 3358 . . 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 13504 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 934    e. wcel 1684    =/= wne 2446    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   |^|cint 3862   ` cfv 5255  Moorecmre 13484
This theorem is referenced by:  submrc  13530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-mre 13488
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