MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  submre Structured version   Unicode version

Theorem submre 14876
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 3724 . . 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 461 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  A  e.  C )
4 pwidg 4029 . . . 4  |-  ( A  e.  C  ->  A  e.  ~P A )
54adantl 466 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  A  e.  ~P A )
63, 5elind 3693 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  A  e.  ( C  i^i  ~P A ) )
7 simp1l 1020 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  C  e.  (Moore `  X )
)
8 inss1 3723 . . . . . 6  |-  ( C  i^i  ~P A ) 
C_  C
9 sstr 3517 . . . . . 6  |-  ( ( x  C_  ( C  i^i  ~P A )  /\  ( C  i^i  ~P A
)  C_  C )  ->  x  C_  C )
108, 9mpan2 671 . . . . 5  |-  ( x 
C_  ( C  i^i  ~P A )  ->  x  C_  C )
11103ad2ant2 1018 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  C_  C )
12 simp3 998 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  =/=  (/) )
13 mreintcl 14866 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  e.  C )
147, 11, 12, 13syl3anc 1228 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  e.  C )
15 sstr 3517 . . . . . . . 8  |-  ( ( x  C_  ( C  i^i  ~P A )  /\  ( C  i^i  ~P A
)  C_  ~P A
)  ->  x  C_  ~P A )
161, 15mpan2 671 . . . . . . 7  |-  ( x 
C_  ( C  i^i  ~P A )  ->  x  C_ 
~P A )
17163ad2ant2 1018 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  C_ 
~P A )
18 intssuni2 4313 . . . . . 6  |-  ( ( x  C_  ~P A  /\  x  =/=  (/) )  ->  |^| x  C_  U. ~P A )
1917, 12, 18syl2anc 661 . . . . 5  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  C_ 
U. ~P A )
20 unipw 4703 . . . . 5  |-  U. ~P A  =  A
2119, 20syl6sseq 3555 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  C_  A )
22 elpw2g 4616 . . . . . 6  |-  ( A  e.  C  ->  ( |^| x  e.  ~P A 
<-> 
|^| x  C_  A
) )
2322adantl 466 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  ( |^| x  e.  ~P A 
<-> 
|^| x  C_  A
) )
24233ad2ant1 1017 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  ( |^| x  e.  ~P A 
<-> 
|^| x  C_  A
) )
2521, 24mpbird 232 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  e.  ~P A )
2614, 25elind 3693 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  e.  ( C  i^i  ~P A ) )
272, 6, 26ismred 14873 1  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  ( C  i^i  ~P A )  e.  (Moore `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767    =/= wne 2662    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   U.cuni 4251   |^|cint 4288   ` cfv 5594  Moorecmre 14853
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-mre 14857
This theorem is referenced by:  submrc  14899
  Copyright terms: Public domain W3C validator