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Theorem mrcuni 14661
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 457 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  C  e.  (Moore `  X
) )
2 simpll 753 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  C  e.  (Moore `  X ) )
3 ssel2 3449 . . . . . . . . 9  |-  ( ( U  C_  ~P X  /\  s  e.  U
)  ->  s  e.  ~P X )
43elpwid 3968 . . . . . . . 8  |-  ( ( U  C_  ~P X  /\  s  e.  U
)  ->  s  C_  X )
54adantll 713 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  s  C_  X )
6 mrcfval.f . . . . . . . 8  |-  F  =  (mrCls `  C )
76mrcssid 14657 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  s  C_  X )  ->  s  C_  ( F `  s
) )
82, 5, 7syl2anc 661 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  s  C_  ( F `  s ) )
96mrcf 14649 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
10 ffun 5659 . . . . . . . . . . 11  |-  ( F : ~P X --> C  ->  Fun  F )
119, 10syl 16 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  Fun  F )
1211adantr 465 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  Fun  F )
13 fdm 5661 . . . . . . . . . . . 12  |-  ( F : ~P X --> C  ->  dom  F  =  ~P X
)
149, 13syl 16 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  dom  F  =  ~P X )
1514sseq2d 3482 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( U  C_ 
dom  F  <->  U  C_  ~P X
) )
1615biimpar 485 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U  C_  dom  F )
17 funfvima2 6052 . . . . . . . . 9  |-  ( ( Fun  F  /\  U  C_ 
dom  F )  -> 
( s  e.  U  ->  ( F `  s
)  e.  ( F
" U ) ) )
1812, 16, 17syl2anc 661 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( s  e.  U  ->  ( F `  s
)  e.  ( F
" U ) ) )
1918imp 429 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  ( F `  s
)  e.  ( F
" U ) )
20 elssuni 4219 . . . . . . 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 3464 . . . . 5  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  s  C_  U. ( F " U ) )
2322ralrimiva 2822 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. s  e.  U  s  C_  U. ( F
" U ) )
24 unissb 4221 . . . 4  |-  ( U. U  C_  U. ( F
" U )  <->  A. s  e.  U  s  C_  U. ( F " U
) )
2523, 24sylibr 212 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. U  C_  U. ( F " U ) )
266mrcssv 14654 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  ( F `  x )  C_  X
)
2726adantr 465 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  x
)  C_  X )
2827ralrimivw 2823 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. x  e.  U  ( F `  x ) 
C_  X )
29 ffn 5657 . . . . . . 7  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
309, 29syl 16 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  F  Fn  ~P X )
31 sseq1 3475 . . . . . . 7  |-  ( s  =  ( F `  x )  ->  (
s  C_  X  <->  ( F `  x )  C_  X
) )
3231ralima 6056 . . . . . 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 471 . . . . 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 232 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. s  e.  ( F " U ) s 
C_  X )
35 unissb 4221 . . . 4  |-  ( U. ( F " U ) 
C_  X  <->  A. s  e.  ( F " U
) s  C_  X
)
3634, 35sylibr 212 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. ( F " U
)  C_  X )
376mrcss 14656 . . 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 1219 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. U )  C_  ( F `  U. ( F
" U ) ) )
39 simpll 753 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  C  e.  (Moore `  X ) )
40 elssuni 4219 . . . . . . . . 9  |-  ( x  e.  U  ->  x  C_ 
U. U )
4140adantl 466 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  x  C_  U. U )
42 sspwuni 4354 . . . . . . . . . . 11  |-  ( U 
C_  ~P X  <->  U. U  C_  X )
4342biimpi 194 . . . . . . . . . 10  |-  ( U 
C_  ~P X  ->  U. U  C_  X )
4443adantl 466 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. U  C_  X )
4544adantr 465 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  U. U  C_  X
)
466mrcss 14656 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  x  C_ 
U. U  /\  U. U  C_  X )  -> 
( F `  x
)  C_  ( F `  U. U ) )
4739, 41, 45, 46syl3anc 1219 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  ( F `  x
)  C_  ( F `  U. U ) )
4847ralrimiva 2822 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. x  e.  U  ( F `  x ) 
C_  ( F `  U. U ) )
49 sseq1 3475 . . . . . . . 8  |-  ( s  =  ( F `  x )  ->  (
s  C_  ( F `  U. U )  <->  ( F `  x )  C_  ( F `  U. U ) ) )
5049ralima 6056 . . . . . . 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 471 . . . . . 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 232 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. s  e.  ( F " U ) s 
C_  ( F `  U. U ) )
53 unissb 4221 . . . . 5  |-  ( U. ( F " U ) 
C_  ( F `  U. U )  <->  A. s  e.  ( F " U
) s  C_  ( F `  U. U ) )
5452, 53sylibr 212 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. ( F " U
)  C_  ( F `  U. U ) )
556mrcssv 14654 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  ( F `  U. U )  C_  X )
5655adantr 465 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. U )  C_  X
)
576mrcss 14656 . . . 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 1219 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. ( F " U ) )  C_  ( F `  ( F `  U. U ) ) )
596mrcidm 14659 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  X )  -> 
( F `  ( F `  U. U ) )  =  ( F `
 U. U ) )
601, 44, 59syl2anc 661 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  ( F `  U. U ) )  =  ( F `
 U. U ) )
6158, 60sseqtrd 3490 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. ( F " U ) )  C_  ( F `  U. U ) )
6238, 61eqssd 3471 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. U )  =  ( F `  U. ( F " U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795    C_ wss 3426   ~Pcpw 3958   U.cuni 4189   dom cdm 4938   "cima 4941   Fun wfun 5510    Fn wfn 5511   -->wf 5512   ` cfv 5516  Moorecmre 14622  mrClscmrc 14623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-int 4227  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-fv 5524  df-mre 14626  df-mrc 14627
This theorem is referenced by:  mrcun  14662  isacs4lem  15440
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