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Theorem mrcuni 13851
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 445 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  C  e.  (Moore `  X
) )
2 simpll 732 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  C  e.  (Moore `  X ) )
3 ssel2 3345 . . . . . . . . 9  |-  ( ( U  C_  ~P X  /\  s  e.  U
)  ->  s  e.  ~P X )
43elpwid 3810 . . . . . . . 8  |-  ( ( U  C_  ~P X  /\  s  e.  U
)  ->  s  C_  X )
54adantll 696 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  s  C_  X )
6 mrcfval.f . . . . . . . 8  |-  F  =  (mrCls `  C )
76mrcssid 13847 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  s  C_  X )  ->  s  C_  ( F `  s
) )
82, 5, 7syl2anc 644 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  s  C_  ( F `  s ) )
96mrcf 13839 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
10 ffun 5596 . . . . . . . . . . 11  |-  ( F : ~P X --> C  ->  Fun  F )
119, 10syl 16 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  Fun  F )
1211adantr 453 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  Fun  F )
13 fdm 5598 . . . . . . . . . . . 12  |-  ( F : ~P X --> C  ->  dom  F  =  ~P X
)
149, 13syl 16 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  dom  F  =  ~P X )
1514sseq2d 3378 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( U  C_ 
dom  F  <->  U  C_  ~P X
) )
1615biimpar 473 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U  C_  dom  F )
17 funfvima2 5977 . . . . . . . . 9  |-  ( ( Fun  F  /\  U  C_ 
dom  F )  -> 
( s  e.  U  ->  ( F `  s
)  e.  ( F
" U ) ) )
1812, 16, 17syl2anc 644 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( s  e.  U  ->  ( F `  s
)  e.  ( F
" U ) ) )
1918imp 420 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  ( F `  s
)  e.  ( F
" U ) )
20 elssuni 4045 . . . . . . 7  |-  ( ( F `  s )  e.  ( F " U )  ->  ( F `  s )  C_ 
U. ( F " U ) )
2119, 20syl 16 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  ( F `  s
)  C_  U. ( F " U ) )
228, 21sstrd 3360 . . . . 5  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  s  C_  U. ( F " U ) )
2322ralrimiva 2791 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. s  e.  U  s  C_  U. ( F
" U ) )
24 unissb 4047 . . . 4  |-  ( U. U  C_  U. ( F
" U )  <->  A. s  e.  U  s  C_  U. ( F " U
) )
2523, 24sylibr 205 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. U  C_  U. ( F " U ) )
266mrcssv 13844 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  ( F `  x )  C_  X
)
2726adantr 453 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  x
)  C_  X )
2827ralrimivw 2792 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. x  e.  U  ( F `  x ) 
C_  X )
29 ffn 5594 . . . . . . 7  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
309, 29syl 16 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  F  Fn  ~P X )
31 sseq1 3371 . . . . . . 7  |-  ( s  =  ( F `  x )  ->  (
s  C_  X  <->  ( F `  x )  C_  X
) )
3231ralima 5981 . . . . . 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 ) )
3330, 32sylan 459 . . . . 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 ) )
3428, 33mpbird 225 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. s  e.  ( F " U ) s 
C_  X )
35 unissb 4047 . . . 4  |-  ( U. ( F " U ) 
C_  X  <->  A. s  e.  ( F " U
) s  C_  X
)
3634, 35sylibr 205 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. ( F " U
)  C_  X )
376mrcss 13846 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  U. ( F
" U )  /\  U. ( F " U
)  C_  X )  ->  ( F `  U. U )  C_  ( F `  U. ( F
" U ) ) )
381, 25, 36, 37syl3anc 1185 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. U )  C_  ( F `  U. ( F
" U ) ) )
39 simpll 732 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  C  e.  (Moore `  X ) )
40 elssuni 4045 . . . . . . . . 9  |-  ( x  e.  U  ->  x  C_ 
U. U )
4140adantl 454 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  x  C_  U. U )
42 sspwuni 4179 . . . . . . . . . . 11  |-  ( U 
C_  ~P X  <->  U. U  C_  X )
4342biimpi 188 . . . . . . . . . 10  |-  ( U 
C_  ~P X  ->  U. U  C_  X )
4443adantl 454 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. U  C_  X )
4544adantr 453 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  U. U  C_  X
)
466mrcss 13846 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  x  C_ 
U. U  /\  U. U  C_  X )  -> 
( F `  x
)  C_  ( F `  U. U ) )
4739, 41, 45, 46syl3anc 1185 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  ( F `  x
)  C_  ( F `  U. U ) )
4847ralrimiva 2791 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. x  e.  U  ( F `  x ) 
C_  ( F `  U. U ) )
49 sseq1 3371 . . . . . . . 8  |-  ( s  =  ( F `  x )  ->  (
s  C_  ( F `  U. U )  <->  ( F `  x )  C_  ( F `  U. U ) ) )
5049ralima 5981 . . . . . . 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 ) ) )
5130, 50sylan 459 . . . . . 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 ) ) )
5248, 51mpbird 225 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. s  e.  ( F " U ) s 
C_  ( F `  U. U ) )
53 unissb 4047 . . . . 5  |-  ( U. ( F " U ) 
C_  ( F `  U. U )  <->  A. s  e.  ( F " U
) s  C_  ( F `  U. U ) )
5452, 53sylibr 205 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. ( F " U
)  C_  ( F `  U. U ) )
556mrcssv 13844 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  ( F `  U. U )  C_  X )
5655adantr 453 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. U )  C_  X
)
576mrcss 13846 . . . 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 ) ) )
581, 54, 56, 57syl3anc 1185 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. ( F " U ) )  C_  ( F `  ( F `  U. U ) ) )
596mrcidm 13849 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  X )  -> 
( F `  ( F `  U. U ) )  =  ( F `
 U. U ) )
601, 44, 59syl2anc 644 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  ( F `  U. U ) )  =  ( F `
 U. U ) )
6158, 60sseqtrd 3386 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. ( F " U ) )  C_  ( F `  U. U ) )
6238, 61eqssd 3367 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   dom cdm 4881   "cima 4884   Fun wfun 5451    Fn wfn 5452   -->wf 5453   ` cfv 5457  Moorecmre 13812  mrClscmrc 13813
This theorem is referenced by:  mrcun  13852  isacs4lem  14599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-mre 13816  df-mrc 13817
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