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Theorem mrcuni 13523
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 3175 . . . . . . . . 9  |-  ( ( U  C_  ~P X  /\  s  e.  U
)  ->  s  e.  ~P X )
4 elpwi 3633 . . . . . . . . 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 13519 . . . . . . 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 13511 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
11 ffun 5391 . . . . . . . . . . 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 5393 . . . . . . . . . . . 12  |-  ( F : ~P X --> C  ->  dom  F  =  ~P X
)
1510, 14syl 15 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  dom  F  =  ~P X )
1615sseq2d 3206 . . . . . . . . . 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 5754 . . . . . . . . 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 3855 . . . . . . 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 3189 . . . . 5  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  s  C_  U. ( F " U ) )
2423ralrimiva 2626 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. s  e.  U  s  C_  U. ( F
" U ) )
25 unissb 3857 . . . 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 13516 . . . . . . 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 2627 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. x  e.  U  ( F `  x ) 
C_  X )
30 ffn 5389 . . . . . . 7  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
3110, 30syl 15 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  F  Fn  ~P X )
32 sseq1 3199 . . . . . . 7  |-  ( s  =  ( F `  x )  ->  (
s  C_  X  <->  ( F `  x )  C_  X
) )
3332ralima 5758 . . . . . 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 3857 . . . 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 13518 . . 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 3855 . . . . . . . . 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 3987 . . . . . . . . . . 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 13518 . . . . . . . 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 2626 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. x  e.  U  ( F `  x ) 
C_  ( F `  U. U ) )
50 sseq1 3199 . . . . . . . 8  |-  ( s  =  ( F `  x )  ->  (
s  C_  ( F `  U. U )  <->  ( F `  x )  C_  ( F `  U. U ) ) )
5150ralima 5758 . . . . . . 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 3857 . . . . 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 13516 . . . . 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 13518 . . . 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 13521 . . . 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 3214 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. ( F " U ) )  C_  ( F `  U. U ) )
6339, 62eqssd 3196 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 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   dom cdm 4689   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  Moorecmre 13484  mrClscmrc 13485
This theorem is referenced by:  mrcun  13524  isacs4lem  14271
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  ax-un 4512
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-csb 3082  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-iun 3907  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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-mre 13488  df-mrc 13489
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