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Theorem mrcuni 13834
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 444 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  C  e.  (Moore `  X
) )
2 simpll 731 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  C  e.  (Moore `  X ) )
3 ssel2 3335 . . . . . . . . 9  |-  ( ( U  C_  ~P X  /\  s  e.  U
)  ->  s  e.  ~P X )
43elpwid 3800 . . . . . . . 8  |-  ( ( U  C_  ~P X  /\  s  e.  U
)  ->  s  C_  X )
54adantll 695 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  s  C_  X )
6 mrcfval.f . . . . . . . 8  |-  F  =  (mrCls `  C )
76mrcssid 13830 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  s  C_  X )  ->  s  C_  ( F `  s
) )
82, 5, 7syl2anc 643 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  s  C_  ( F `  s ) )
96mrcf 13822 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
10 ffun 5584 . . . . . . . . . . 11  |-  ( F : ~P X --> C  ->  Fun  F )
119, 10syl 16 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  Fun  F )
1211adantr 452 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  Fun  F )
13 fdm 5586 . . . . . . . . . . . 12  |-  ( F : ~P X --> C  ->  dom  F  =  ~P X
)
149, 13syl 16 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  dom  F  =  ~P X )
1514sseq2d 3368 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( U  C_ 
dom  F  <->  U  C_  ~P X
) )
1615biimpar 472 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U  C_  dom  F )
17 funfvima2 5965 . . . . . . . . 9  |-  ( ( Fun  F  /\  U  C_ 
dom  F )  -> 
( s  e.  U  ->  ( F `  s
)  e.  ( F
" U ) ) )
1812, 16, 17syl2anc 643 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( s  e.  U  ->  ( F `  s
)  e.  ( F
" U ) ) )
1918imp 419 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  ( F `  s
)  e.  ( F
" U ) )
20 elssuni 4035 . . . . . . 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 3350 . . . . 5  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  s  e.  U )  ->  s  C_  U. ( F " U ) )
2322ralrimiva 2781 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. s  e.  U  s  C_  U. ( F
" U ) )
24 unissb 4037 . . . 4  |-  ( U. U  C_  U. ( F
" U )  <->  A. s  e.  U  s  C_  U. ( F " U
) )
2523, 24sylibr 204 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. U  C_  U. ( F " U ) )
266mrcssv 13827 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  ( F `  x )  C_  X
)
2726adantr 452 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  x
)  C_  X )
2827ralrimivw 2782 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. x  e.  U  ( F `  x ) 
C_  X )
29 ffn 5582 . . . . . . 7  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
309, 29syl 16 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  F  Fn  ~P X )
31 sseq1 3361 . . . . . . 7  |-  ( s  =  ( F `  x )  ->  (
s  C_  X  <->  ( F `  x )  C_  X
) )
3231ralima 5969 . . . . . 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 458 . . . . 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 224 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. s  e.  ( F " U ) s 
C_  X )
35 unissb 4037 . . . 4  |-  ( U. ( F " U ) 
C_  X  <->  A. s  e.  ( F " U
) s  C_  X
)
3634, 35sylibr 204 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. ( F " U
)  C_  X )
376mrcss 13829 . . 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 1184 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. U )  C_  ( F `  U. ( F
" U ) ) )
39 simpll 731 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  C  e.  (Moore `  X ) )
40 elssuni 4035 . . . . . . . . 9  |-  ( x  e.  U  ->  x  C_ 
U. U )
4140adantl 453 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  x  C_  U. U )
42 sspwuni 4168 . . . . . . . . . . 11  |-  ( U 
C_  ~P X  <->  U. U  C_  X )
4342biimpi 187 . . . . . . . . . 10  |-  ( U 
C_  ~P X  ->  U. U  C_  X )
4443adantl 453 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. U  C_  X )
4544adantr 452 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  U. U  C_  X
)
466mrcss 13829 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  x  C_ 
U. U  /\  U. U  C_  X )  -> 
( F `  x
)  C_  ( F `  U. U ) )
4739, 41, 45, 46syl3anc 1184 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  /\  x  e.  U )  ->  ( F `  x
)  C_  ( F `  U. U ) )
4847ralrimiva 2781 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. x  e.  U  ( F `  x ) 
C_  ( F `  U. U ) )
49 sseq1 3361 . . . . . . . 8  |-  ( s  =  ( F `  x )  ->  (
s  C_  ( F `  U. U )  <->  ( F `  x )  C_  ( F `  U. U ) ) )
5049ralima 5969 . . . . . . 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 458 . . . . . 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 224 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  A. s  e.  ( F " U ) s 
C_  ( F `  U. U ) )
53 unissb 4037 . . . . 5  |-  ( U. ( F " U ) 
C_  ( F `  U. U )  <->  A. s  e.  ( F " U
) s  C_  ( F `  U. U ) )
5452, 53sylibr 204 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  ->  U. ( F " U
)  C_  ( F `  U. U ) )
556mrcssv 13827 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  ( F `  U. U )  C_  X )
5655adantr 452 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. U )  C_  X
)
576mrcss 13829 . . . 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 1184 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. ( F " U ) )  C_  ( F `  ( F `  U. U ) ) )
596mrcidm 13832 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  X )  -> 
( F `  ( F `  U. U ) )  =  ( F `
 U. U ) )
601, 44, 59syl2anc 643 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  ( F `  U. U ) )  =  ( F `
 U. U ) )
6158, 60sseqtrd 3376 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_ 
~P X )  -> 
( F `  U. ( F " U ) )  C_  ( F `  U. U ) )
6238, 61eqssd 3357 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   dom cdm 4869   "cima 4872   Fun wfun 5439    Fn wfn 5440   -->wf 5441   ` cfv 5445  Moorecmre 13795  mrClscmrc 13796
This theorem is referenced by:  mrcun  13835  isacs4lem  14582
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-fv 5453  df-mre 13799  df-mrc 13800
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