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Theorem submre 14564
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 3592 . . 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 3894 . . . 4  |-  ( A  e.  C  ->  A  e.  ~P A )
54adantl 466 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  A  e.  ~P A )
63, 5elind 3561 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  A  e.  ( C  i^i  ~P A ) )
7 simp1l 1012 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  C  e.  (Moore `  X )
)
8 inss1 3591 . . . . . 6  |-  ( C  i^i  ~P A ) 
C_  C
9 sstr 3385 . . . . . 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 1010 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  C_  C )
12 simp3 990 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  =/=  (/) )
13 mreintcl 14554 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  e.  C )
147, 11, 12, 13syl3anc 1218 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  e.  C )
15 sstr 3385 . . . . . . . 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 1010 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  x  C_ 
~P A )
18 intssuni2 4174 . . . . . 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 4563 . . . . 5  |-  U. ~P A  =  A
2119, 20syl6sseq 3423 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A  e.  C )  /\  x  C_  ( C  i^i  ~P A )  /\  x  =/=  (/) )  ->  |^| x  C_  A )
22 elpw2g 4476 . . . . . 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 1009 . . . 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 3561 . 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 14561 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 965    e. wcel 1756    =/= wne 2620    i^i cin 3348    C_ wss 3349   (/)c0 3658   ~Pcpw 3881   U.cuni 4112   |^|cint 4149   ` cfv 5439  Moorecmre 14541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-int 4150  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-iota 5402  df-fun 5441  df-fv 5447  df-mre 14545
This theorem is referenced by:  submrc  14587
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