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Theorem mrcun 13839
Description: Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcun  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  ( U  u.  V ) )  =  ( F `  (
( F `  U
)  u.  ( F `
 V ) ) ) )

Proof of Theorem mrcun
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  C  e.  (Moore `  X )
)
2 mre1cl 13811 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
3 elpw2g 4355 . . . . . . 7  |-  ( X  e.  C  ->  ( U  e.  ~P X  <->  U 
C_  X ) )
42, 3syl 16 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  ( U  e.  ~P X  <->  U  C_  X
) )
54biimpar 472 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  e.  ~P X )
653adant3 977 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U  e.  ~P X )
7 elpw2g 4355 . . . . . . 7  |-  ( X  e.  C  ->  ( V  e.  ~P X  <->  V 
C_  X ) )
82, 7syl 16 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  ( V  e.  ~P X  <->  V  C_  X
) )
98biimpar 472 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  V  C_  X )  ->  V  e.  ~P X )
1093adant2 976 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  V  e.  ~P X )
11 prssi 3946 . . . 4  |-  ( ( U  e.  ~P X  /\  V  e.  ~P X )  ->  { U ,  V }  C_  ~P X )
126, 10, 11syl2anc 643 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  { U ,  V }  C_  ~P X )
13 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
1413mrcuni 13838 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  { U ,  V }  C_ 
~P X )  -> 
( F `  U. { U ,  V }
)  =  ( F `
 U. ( F
" { U ,  V } ) ) )
151, 12, 14syl2anc 643 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  U. { U ,  V } )  =  ( F `  U. ( F " { U ,  V } ) ) )
16 uniprg 4022 . . . 4  |-  ( ( U  e.  ~P X  /\  V  e.  ~P X )  ->  U. { U ,  V }  =  ( U  u.  V ) )
176, 10, 16syl2anc 643 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U. { U ,  V }  =  ( U  u.  V ) )
1817fveq2d 5724 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  U. { U ,  V } )  =  ( F `  ( U  u.  V )
) )
1913mrcf 13826 . . . . . . . 8  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
20 ffn 5583 . . . . . . . 8  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
2119, 20syl 16 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  F  Fn  ~P X )
22213ad2ant1 978 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  F  Fn  ~P X )
23 fnimapr 5779 . . . . . 6  |-  ( ( F  Fn  ~P X  /\  U  e.  ~P X  /\  V  e.  ~P X )  ->  ( F " { U ,  V } )  =  {
( F `  U
) ,  ( F `
 V ) } )
2422, 6, 10, 23syl3anc 1184 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F " { U ,  V } )  =  {
( F `  U
) ,  ( F `
 V ) } )
2524unieqd 4018 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U. ( F " { U ,  V } )  =  U. { ( F `  U ) ,  ( F `  V ) } )
26 fvex 5734 . . . . 5  |-  ( F `
 U )  e. 
_V
27 fvex 5734 . . . . 5  |-  ( F `
 V )  e. 
_V
2826, 27unipr 4021 . . . 4  |-  U. {
( F `  U
) ,  ( F `
 V ) }  =  ( ( F `
 U )  u.  ( F `  V
) )
2925, 28syl6eq 2483 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U. ( F " { U ,  V } )  =  ( ( F `  U
)  u.  ( F `
 V ) ) )
3029fveq2d 5724 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  U. ( F
" { U ,  V } ) )  =  ( F `  (
( F `  U
)  u.  ( F `
 V ) ) ) )
3115, 18, 303eqtr3d 2475 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  ( U  u.  V ) )  =  ( F `  (
( F `  U
)  u.  ( F `
 V ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725    u. cun 3310    C_ wss 3312   ~Pcpw 3791   {cpr 3807   U.cuni 4007   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  Moorecmre 13799  mrClscmrc 13800
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 4693
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 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-mre 13803  df-mrc 13804
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