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

Theorem mrcuni 13539
Description: Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcuni  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. U )  =  ( F `  U. ( F " U ) ) )

Proof of Theorem mrcuni
Dummy variables  x  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  C  e.  (Moore `  X
) )
2 simpll 730 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  C  e.  (Moore `  X ) )
3 ssel2 3188 . . . . . . . . 9  |-  ( ( U  C_  ~P X  /\  s  e.  U
)  ->  s  e.  ~P X )
4 elpwi 3646 . . . . . . . . 9  |-  ( s  e.  ~P X  -> 
s  C_  X )
53, 4syl 15 . . . . . . . 8  |-  ( ( U  C_  ~P X  /\  s  e.  U
)  ->  s  C_  X )
65adantll 694 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  s  C_  X )
7 mrcfval.f . . . . . . . 8  |-  F  =  (mrCls `  C )
87mrcssid 13535 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  s  C_  X )  ->  s  C_  ( F `  s
) )
92, 6, 8syl2anc 642 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  s  C_  ( F `  s ) )
107mrcf 13527 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
11 ffun 5407 . . . . . . . . . . 11  |-  ( F : ~P X --> C  ->  Fun  F )
1210, 11syl 15 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  Fun  F )
1312adantr 451 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  Fun  F )
14 fdm 5409 . . . . . . . . . . . 12  |-  ( F : ~P X --> C  ->  dom  F  =  ~P X
)
1510, 14syl 15 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  dom  F  =  ~P X )
1615sseq2d 3219 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( U  C_ 
dom  F  <->  U  C_  ~P X
) )
1716biimpar 471 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U  C_  dom  F )
18 funfvima2 5770 . . . . . . . . 9  |-  ( ( Fun  F  /\  U  C_ 
dom  F )  -> 
( s  e.  U  ->  ( F `  s
)  e.  ( F
" U ) ) )
1913, 17, 18syl2anc 642 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( s  e.  U  ->  ( F `  s
)  e.  ( F
" U ) ) )
2019imp 418 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  ( F `  s
)  e.  ( F
" U ) )
21 elssuni 3871 . . . . . . 7  |-  ( ( F `  s )  e.  ( F " U )  ->  ( F `  s )  C_ 
U. ( F " U ) )
2220, 21syl 15 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  ( F `  s
)  C_  U. ( F " U ) )
239, 22sstrd 3202 . . . . 5  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  s  C_  U. ( F " U ) )
2423ralrimiva 2639 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. s  e.  U  s  C_  U. ( F
" U ) )
25 unissb 3873 . . . 4  |-  ( U. U  C_  U. ( F
" U )  <->  A. s  e.  U  s  C_  U. ( F " U
) )
2624, 25sylibr 203 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. U  C_  U. ( F " U ) )
277mrcssv 13532 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  ( F `  x )  C_  X
)
2827adantr 451 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  x
)  C_  X )
2928ralrimivw 2640 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. x  e.  U  ( F `  x ) 
C_  X )
30 ffn 5405 . . . . . . 7  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
3110, 30syl 15 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  F  Fn  ~P X )
32 sseq1 3212 . . . . . . 7  |-  ( s  =  ( F `  x )  ->  (
s  C_  X  <->  ( F `  x )  C_  X
) )
3332ralima 5774 . . . . . 6  |-  ( ( F  Fn  ~P X  /\  U  C_  ~P X
)  ->  ( A. s  e.  ( F " U ) s  C_  X 
<-> 
A. x  e.  U  ( F `  x ) 
C_  X ) )
3431, 33sylan 457 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( A. s  e.  ( F " U
) s  C_  X  <->  A. x  e.  U  ( F `  x ) 
C_  X ) )
3529, 34mpbird 223 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. s  e.  ( F " U ) s 
C_  X )
36 unissb 3873 . . . 4  |-  ( U. ( F " U ) 
C_  X  <->  A. s  e.  ( F " U
) s  C_  X
)
3735, 36sylibr 203 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. ( F " U
)  C_  X )
387mrcss 13534 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  U. ( F
" U )  /\  U. ( F " U
)  C_  X )  ->  ( F `  U. U )  C_  ( F `  U. ( F
" U ) ) )
391, 26, 37, 38syl3anc 1182 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. U )  C_  ( F `  U. ( F
" U ) ) )
40 simpll 730 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  C  e.  (Moore `  X ) )
41 elssuni 3871 . . . . . . . . 9  |-  ( x  e.  U  ->  x  C_ 
U. U )
4241adantl 452 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  x  C_  U. U )
43 sspwuni 4003 . . . . . . . . . . 11  |-  ( U 
C_  ~P X  <->  U. U  C_  X )
4443biimpi 186 . . . . . . . . . 10  |-  ( U 
C_  ~P X  ->  U. U  C_  X )
4544adantl 452 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. U  C_  X )
4645adantr 451 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  U. U  C_  X
)
477mrcss 13534 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  x  C_ 
U. U  /\  U. U  C_  X )  -> 
( F `  x
)  C_  ( F `  U. U ) )
4840, 42, 46, 47syl3anc 1182 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  ( F `  x
)  C_  ( F `  U. U ) )
4948ralrimiva 2639 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. x  e.  U  ( F `  x ) 
C_  ( F `  U. U ) )
50 sseq1 3212 . . . . . . . 8  |-  ( s  =  ( F `  x )  ->  (
s  C_  ( F `  U. U )  <->  ( F `  x )  C_  ( F `  U. U ) ) )
5150ralima 5774 . . . . . . 7  |-  ( ( F  Fn  ~P X  /\  U  C_  ~P X
)  ->  ( A. s  e.  ( F " U ) s  C_  ( F `  U. U
)  <->  A. x  e.  U  ( F `  x ) 
C_  ( F `  U. U ) ) )
5231, 51sylan 457 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( A. s  e.  ( F " U
) s  C_  ( F `  U. U )  <->  A. x  e.  U  ( F `  x ) 
C_  ( F `  U. U ) ) )
5349, 52mpbird 223 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. s  e.  ( F " U ) s 
C_  ( F `  U. U ) )
54 unissb 3873 . . . . 5  |-  ( U. ( F " U ) 
C_  ( F `  U. U )  <->  A. s  e.  ( F " U
) s  C_  ( F `  U. U ) )
5553, 54sylibr 203 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. ( F " U
)  C_  ( F `  U. U ) )
567mrcssv 13532 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  ( F `  U. U )  C_  X )
5756adantr 451 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. U )  C_  X
)
587mrcss 13534 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U. ( F " U ) 
C_  ( F `  U. U )  /\  ( F `  U. U ) 
C_  X )  -> 
( F `  U. ( F " U ) )  C_  ( F `  ( F `  U. U ) ) )
591, 55, 57, 58syl3anc 1182 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. ( F " U ) )  C_  ( F `  ( F `  U. U ) ) )
607mrcidm 13537 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  X )  -> 
( F `  ( F `  U. U ) )  =  ( F `
 U. U ) )
611, 45, 60syl2anc 642 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  ( F `  U. U ) )  =  ( F `
 U. U ) )
6259, 61sseqtrd 3227 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. ( F " U ) )  C_  ( F `  U. U ) )
6339, 62eqssd 3209 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. U )  =  ( F `  U. ( F " U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   dom cdm 4705   "cima 4708   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271  Moorecmre 13500  mrClscmrc 13501
This theorem is referenced by:  mrcun  13540  isacs4lem  14287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-mre 13504  df-mrc 13505
  Copyright terms: Public domain W3C validator