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Theorem ismrc 35455
Description: A function is a Moore closure operator iff it satisfies mrcssid 15466, mrcss 15465, and mrcidm 15468. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
ismrc  |-  ( F  e.  (mrCls " (Moore `  B ) )  <->  ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y ( ( x  C_  B  /\  y  C_  x )  ->  ( x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) ) )
Distinct variable groups:    x, F, y    x, B, y

Proof of Theorem ismrc
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnmrc 15456 . . . . 5  |- mrCls  Fn  U. ran Moore
2 fnfun 5634 . . . . 5  |-  (mrCls  Fn  U.
ran Moore  ->  Fun mrCls )
31, 2ax-mp 5 . . . 4  |-  Fun mrCls
4 fvelima 5877 . . . 4  |-  ( ( Fun mrCls  /\  F  e.  (mrCls " (Moore `  B )
) )  ->  E. z  e.  (Moore `  B )
(mrCls `  z )  =  F )
53, 4mpan 674 . . 3  |-  ( F  e.  (mrCls " (Moore `  B ) )  ->  E. z  e.  (Moore `  B ) (mrCls `  z )  =  F )
6 elfvex 5852 . . . . . 6  |-  ( z  e.  (Moore `  B
)  ->  B  e.  _V )
7 eqid 2428 . . . . . . . 8  |-  (mrCls `  z )  =  (mrCls `  z )
87mrcf 15458 . . . . . . 7  |-  ( z  e.  (Moore `  B
)  ->  (mrCls `  z
) : ~P B --> z )
9 mresspw 15441 . . . . . . 7  |-  ( z  e.  (Moore `  B
)  ->  z  C_  ~P B )
108, 9fssd 5698 . . . . . 6  |-  ( z  e.  (Moore `  B
)  ->  (mrCls `  z
) : ~P B --> ~P B )
117mrcssid 15466 . . . . . . . . . 10  |-  ( ( z  e.  (Moore `  B )  /\  x  C_  B )  ->  x  C_  ( (mrCls `  z
) `  x )
)
1211adantrr 721 . . . . . . . . 9  |-  ( ( z  e.  (Moore `  B )  /\  (
x  C_  B  /\  y  C_  x ) )  ->  x  C_  (
(mrCls `  z ) `  x ) )
137mrcss 15465 . . . . . . . . . . 11  |-  ( ( z  e.  (Moore `  B )  /\  y  C_  x  /\  x  C_  B )  ->  (
(mrCls `  z ) `  y )  C_  (
(mrCls `  z ) `  x ) )
14133expb 1206 . . . . . . . . . 10  |-  ( ( z  e.  (Moore `  B )  /\  (
y  C_  x  /\  x  C_  B ) )  ->  ( (mrCls `  z ) `  y
)  C_  ( (mrCls `  z ) `  x
) )
1514ancom2s 809 . . . . . . . . 9  |-  ( ( z  e.  (Moore `  B )  /\  (
x  C_  B  /\  y  C_  x ) )  ->  ( (mrCls `  z ) `  y
)  C_  ( (mrCls `  z ) `  x
) )
167mrcidm 15468 . . . . . . . . . 10  |-  ( ( z  e.  (Moore `  B )  /\  x  C_  B )  ->  (
(mrCls `  z ) `  ( (mrCls `  z
) `  x )
)  =  ( (mrCls `  z ) `  x
) )
1716adantrr 721 . . . . . . . . 9  |-  ( ( z  e.  (Moore `  B )  /\  (
x  C_  B  /\  y  C_  x ) )  ->  ( (mrCls `  z ) `  (
(mrCls `  z ) `  x ) )  =  ( (mrCls `  z
) `  x )
)
1812, 15, 173jca 1185 . . . . . . . 8  |-  ( ( z  e.  (Moore `  B )  /\  (
x  C_  B  /\  y  C_  x ) )  ->  ( x  C_  ( (mrCls `  z ) `  x )  /\  (
(mrCls `  z ) `  y )  C_  (
(mrCls `  z ) `  x )  /\  (
(mrCls `  z ) `  ( (mrCls `  z
) `  x )
)  =  ( (mrCls `  z ) `  x
) ) )
1918ex 435 . . . . . . 7  |-  ( z  e.  (Moore `  B
)  ->  ( (
x  C_  B  /\  y  C_  x )  -> 
( x  C_  (
(mrCls `  z ) `  x )  /\  (
(mrCls `  z ) `  y )  C_  (
(mrCls `  z ) `  x )  /\  (
(mrCls `  z ) `  ( (mrCls `  z
) `  x )
)  =  ( (mrCls `  z ) `  x
) ) ) )
2019alrimivv 1768 . . . . . 6  |-  ( z  e.  (Moore `  B
)  ->  A. x A. y ( ( x 
C_  B  /\  y  C_  x )  ->  (
x  C_  ( (mrCls `  z ) `  x
)  /\  ( (mrCls `  z ) `  y
)  C_  ( (mrCls `  z ) `  x
)  /\  ( (mrCls `  z ) `  (
(mrCls `  z ) `  x ) )  =  ( (mrCls `  z
) `  x )
) ) )
216, 10, 203jca 1185 . . . . 5  |-  ( z  e.  (Moore `  B
)  ->  ( B  e.  _V  /\  (mrCls `  z ) : ~P B
--> ~P B  /\  A. x A. y ( ( x  C_  B  /\  y  C_  x )  -> 
( x  C_  (
(mrCls `  z ) `  x )  /\  (
(mrCls `  z ) `  y )  C_  (
(mrCls `  z ) `  x )  /\  (
(mrCls `  z ) `  ( (mrCls `  z
) `  x )
)  =  ( (mrCls `  z ) `  x
) ) ) ) )
22 feq1 5671 . . . . . 6  |-  ( (mrCls `  z )  =  F  ->  ( (mrCls `  z ) : ~P B
--> ~P B  <->  F : ~P B --> ~P B ) )
23 fveq1 5824 . . . . . . . . . 10  |-  ( (mrCls `  z )  =  F  ->  ( (mrCls `  z ) `  x
)  =  ( F `
 x ) )
2423sseq2d 3435 . . . . . . . . 9  |-  ( (mrCls `  z )  =  F  ->  ( x  C_  ( (mrCls `  z ) `  x )  <->  x  C_  ( F `  x )
) )
25 fveq1 5824 . . . . . . . . . 10  |-  ( (mrCls `  z )  =  F  ->  ( (mrCls `  z ) `  y
)  =  ( F `
 y ) )
2625, 23sseq12d 3436 . . . . . . . . 9  |-  ( (mrCls `  z )  =  F  ->  ( ( (mrCls `  z ) `  y
)  C_  ( (mrCls `  z ) `  x
)  <->  ( F `  y )  C_  ( F `  x )
) )
27 id 22 . . . . . . . . . . 11  |-  ( (mrCls `  z )  =  F  ->  (mrCls `  z
)  =  F )
2827, 23fveq12d 5831 . . . . . . . . . 10  |-  ( (mrCls `  z )  =  F  ->  ( (mrCls `  z ) `  (
(mrCls `  z ) `  x ) )  =  ( F `  ( F `  x )
) )
2928, 23eqeq12d 2443 . . . . . . . . 9  |-  ( (mrCls `  z )  =  F  ->  ( ( (mrCls `  z ) `  (
(mrCls `  z ) `  x ) )  =  ( (mrCls `  z
) `  x )  <->  ( F `  ( F `
 x ) )  =  ( F `  x ) ) )
3024, 26, 293anbi123d 1335 . . . . . . . 8  |-  ( (mrCls `  z )  =  F  ->  ( ( x 
C_  ( (mrCls `  z ) `  x
)  /\  ( (mrCls `  z ) `  y
)  C_  ( (mrCls `  z ) `  x
)  /\  ( (mrCls `  z ) `  (
(mrCls `  z ) `  x ) )  =  ( (mrCls `  z
) `  x )
)  <->  ( x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) )
3130imbi2d 317 . . . . . . 7  |-  ( (mrCls `  z )  =  F  ->  ( ( ( x  C_  B  /\  y  C_  x )  -> 
( x  C_  (
(mrCls `  z ) `  x )  /\  (
(mrCls `  z ) `  y )  C_  (
(mrCls `  z ) `  x )  /\  (
(mrCls `  z ) `  ( (mrCls `  z
) `  x )
)  =  ( (mrCls `  z ) `  x
) ) )  <->  ( (
x  C_  B  /\  y  C_  x )  -> 
( x  C_  ( F `  x )  /\  ( F `  y
)  C_  ( F `  x )  /\  ( F `  ( F `  x ) )  =  ( F `  x
) ) ) ) )
32312albidv 1763 . . . . . 6  |-  ( (mrCls `  z )  =  F  ->  ( A. x A. y ( ( x 
C_  B  /\  y  C_  x )  ->  (
x  C_  ( (mrCls `  z ) `  x
)  /\  ( (mrCls `  z ) `  y
)  C_  ( (mrCls `  z ) `  x
)  /\  ( (mrCls `  z ) `  (
(mrCls `  z ) `  x ) )  =  ( (mrCls `  z
) `  x )
) )  <->  A. x A. y ( ( x 
C_  B  /\  y  C_  x )  ->  (
x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x
)  /\  ( F `  ( F `  x
) )  =  ( F `  x ) ) ) ) )
3322, 323anbi23d 1338 . . . . 5  |-  ( (mrCls `  z )  =  F  ->  ( ( B  e.  _V  /\  (mrCls `  z ) : ~P B
--> ~P B  /\  A. x A. y ( ( x  C_  B  /\  y  C_  x )  -> 
( x  C_  (
(mrCls `  z ) `  x )  /\  (
(mrCls `  z ) `  y )  C_  (
(mrCls `  z ) `  x )  /\  (
(mrCls `  z ) `  ( (mrCls `  z
) `  x )
)  =  ( (mrCls `  z ) `  x
) ) ) )  <-> 
( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y ( ( x 
C_  B  /\  y  C_  x )  ->  (
x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x
)  /\  ( F `  ( F `  x
) )  =  ( F `  x ) ) ) ) ) )
3421, 33syl5ibcom 223 . . . 4  |-  ( z  e.  (Moore `  B
)  ->  ( (mrCls `  z )  =  F  ->  ( B  e. 
_V  /\  F : ~P B --> ~P B  /\  A. x A. y ( ( x  C_  B  /\  y  C_  x )  ->  ( x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) ) ) )
3534rexlimiv 2850 . . 3  |-  ( E. z  e.  (Moore `  B ) (mrCls `  z )  =  F  ->  ( B  e. 
_V  /\  F : ~P B --> ~P B  /\  A. x A. y ( ( x  C_  B  /\  y  C_  x )  ->  ( x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) ) )
365, 35syl 17 . 2  |-  ( F  e.  (mrCls " (Moore `  B ) )  -> 
( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y ( ( x 
C_  B  /\  y  C_  x )  ->  (
x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x
)  /\  ( F `  ( F `  x
) )  =  ( F `  x ) ) ) ) )
37 simp1 1005 . . . 4  |-  ( ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y
( ( x  C_  B  /\  y  C_  x
)  ->  ( x  C_  ( F `  x
)  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) )  ->  B  e.  _V )
38 simp2 1006 . . . 4  |-  ( ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y
( ( x  C_  B  /\  y  C_  x
)  ->  ( x  C_  ( F `  x
)  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) )  ->  F : ~P B --> ~P B
)
39 ssid 3426 . . . . . . 7  |-  z  C_  z
40 3simpb 1003 . . . . . . . . . . 11  |-  ( ( x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x
)  /\  ( F `  ( F `  x
) )  =  ( F `  x ) )  ->  ( x  C_  ( F `  x
)  /\  ( F `  ( F `  x
) )  =  ( F `  x ) ) )
4140imim2i 16 . . . . . . . . . 10  |-  ( ( ( x  C_  B  /\  y  C_  x )  ->  ( x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) )  ->  ( (
x  C_  B  /\  y  C_  x )  -> 
( x  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) )
42412alimi 1679 . . . . . . . . 9  |-  ( A. x A. y ( ( x  C_  B  /\  y  C_  x )  -> 
( x  C_  ( F `  x )  /\  ( F `  y
)  C_  ( F `  x )  /\  ( F `  ( F `  x ) )  =  ( F `  x
) ) )  ->  A. x A. y ( ( x  C_  B  /\  y  C_  x )  ->  ( x  C_  ( F `  x )  /\  ( F `  ( F `  x ) )  =  ( F `
 x ) ) ) )
43 vex 3025 . . . . . . . . . 10  |-  z  e. 
_V
44 sseq1 3428 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
x  C_  B  <->  z  C_  B ) )
4544adantr 466 . . . . . . . . . . . . 13  |-  ( ( x  =  z  /\  y  =  z )  ->  ( x  C_  B  <->  z 
C_  B ) )
46 sseq12 3430 . . . . . . . . . . . . . 14  |-  ( ( y  =  z  /\  x  =  z )  ->  ( y  C_  x  <->  z 
C_  z ) )
4746ancoms 454 . . . . . . . . . . . . 13  |-  ( ( x  =  z  /\  y  =  z )  ->  ( y  C_  x  <->  z 
C_  z ) )
4845, 47anbi12d 715 . . . . . . . . . . . 12  |-  ( ( x  =  z  /\  y  =  z )  ->  ( ( x  C_  B  /\  y  C_  x
)  <->  ( z  C_  B  /\  z  C_  z
) ) )
49 id 22 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  x  =  z )
50 fveq2 5825 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
5149, 50sseq12d 3436 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
x  C_  ( F `  x )  <->  z  C_  ( F `  z ) ) )
5251adantr 466 . . . . . . . . . . . . 13  |-  ( ( x  =  z  /\  y  =  z )  ->  ( x  C_  ( F `  x )  <->  z 
C_  ( F `  z ) ) )
5350fveq2d 5829 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  z )
) )
5453, 50eqeq12d 2443 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
( F `  ( F `  x )
)  =  ( F `
 x )  <->  ( F `  ( F `  z
) )  =  ( F `  z ) ) )
5554adantr 466 . . . . . . . . . . . . 13  |-  ( ( x  =  z  /\  y  =  z )  ->  ( ( F `  ( F `  x ) )  =  ( F `
 x )  <->  ( F `  ( F `  z
) )  =  ( F `  z ) ) )
5652, 55anbi12d 715 . . . . . . . . . . . 12  |-  ( ( x  =  z  /\  y  =  z )  ->  ( ( x  C_  ( F `  x )  /\  ( F `  ( F `  x ) )  =  ( F `
 x ) )  <-> 
( z  C_  ( F `  z )  /\  ( F `  ( F `  z )
)  =  ( F `
 z ) ) ) )
5748, 56imbi12d 321 . . . . . . . . . . 11  |-  ( ( x  =  z  /\  y  =  z )  ->  ( ( ( x 
C_  B  /\  y  C_  x )  ->  (
x  C_  ( F `  x )  /\  ( F `  ( F `  x ) )  =  ( F `  x
) ) )  <->  ( (
z  C_  B  /\  z  C_  z )  -> 
( z  C_  ( F `  z )  /\  ( F `  ( F `  z )
)  =  ( F `
 z ) ) ) ) )
5857spc2gv 3112 . . . . . . . . . 10  |-  ( ( z  e.  _V  /\  z  e.  _V )  ->  ( A. x A. y ( ( x 
C_  B  /\  y  C_  x )  ->  (
x  C_  ( F `  x )  /\  ( F `  ( F `  x ) )  =  ( F `  x
) ) )  -> 
( ( z  C_  B  /\  z  C_  z
)  ->  ( z  C_  ( F `  z
)  /\  ( F `  ( F `  z
) )  =  ( F `  z ) ) ) ) )
5943, 43, 58mp2an 676 . . . . . . . . 9  |-  ( A. x A. y ( ( x  C_  B  /\  y  C_  x )  -> 
( x  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) )  ->  ( (
z  C_  B  /\  z  C_  z )  -> 
( z  C_  ( F `  z )  /\  ( F `  ( F `  z )
)  =  ( F `
 z ) ) ) )
6042, 59syl 17 . . . . . . . 8  |-  ( A. x A. y ( ( x  C_  B  /\  y  C_  x )  -> 
( x  C_  ( F `  x )  /\  ( F `  y
)  C_  ( F `  x )  /\  ( F `  ( F `  x ) )  =  ( F `  x
) ) )  -> 
( ( z  C_  B  /\  z  C_  z
)  ->  ( z  C_  ( F `  z
)  /\  ( F `  ( F `  z
) )  =  ( F `  z ) ) ) )
61603ad2ant3 1028 . . . . . . 7  |-  ( ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y
( ( x  C_  B  /\  y  C_  x
)  ->  ( x  C_  ( F `  x
)  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) )  ->  (
( z  C_  B  /\  z  C_  z )  ->  ( z  C_  ( F `  z )  /\  ( F `  ( F `  z ) )  =  ( F `
 z ) ) ) )
6239, 61mpan2i 681 . . . . . 6  |-  ( ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y
( ( x  C_  B  /\  y  C_  x
)  ->  ( x  C_  ( F `  x
)  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) )  ->  (
z  C_  B  ->  ( z  C_  ( F `  z )  /\  ( F `  ( F `  z ) )  =  ( F `  z
) ) ) )
6362imp 430 . . . . 5  |-  ( ( ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y ( ( x 
C_  B  /\  y  C_  x )  ->  (
x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x
)  /\  ( F `  ( F `  x
) )  =  ( F `  x ) ) ) )  /\  z  C_  B )  -> 
( z  C_  ( F `  z )  /\  ( F `  ( F `  z )
)  =  ( F `
 z ) ) )
6463simpld 460 . . . 4  |-  ( ( ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y ( ( x 
C_  B  /\  y  C_  x )  ->  (
x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x
)  /\  ( F `  ( F `  x
) )  =  ( F `  x ) ) ) )  /\  z  C_  B )  -> 
z  C_  ( F `  z ) )
65 simp2 1006 . . . . . . . . 9  |-  ( ( x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x
)  /\  ( F `  ( F `  x
) )  =  ( F `  x ) )  ->  ( F `  y )  C_  ( F `  x )
)
6665imim2i 16 . . . . . . . 8  |-  ( ( ( x  C_  B  /\  y  C_  x )  ->  ( x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) )  ->  ( (
x  C_  B  /\  y  C_  x )  -> 
( F `  y
)  C_  ( F `  x ) ) )
67662alimi 1679 . . . . . . 7  |-  ( A. x A. y ( ( x  C_  B  /\  y  C_  x )  -> 
( x  C_  ( F `  x )  /\  ( F `  y
)  C_  ( F `  x )  /\  ( F `  ( F `  x ) )  =  ( F `  x
) ) )  ->  A. x A. y ( ( x  C_  B  /\  y  C_  x )  ->  ( F `  y )  C_  ( F `  x )
) )
68673ad2ant3 1028 . . . . . 6  |-  ( ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y
( ( x  C_  B  /\  y  C_  x
)  ->  ( x  C_  ( F `  x
)  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) )  ->  A. x A. y ( ( x 
C_  B  /\  y  C_  x )  ->  ( F `  y )  C_  ( F `  x
) ) )
69 vex 3025 . . . . . . 7  |-  w  e. 
_V
7044adantr 466 . . . . . . . . . 10  |-  ( ( x  =  z  /\  y  =  w )  ->  ( x  C_  B  <->  z 
C_  B ) )
71 sseq12 3430 . . . . . . . . . . 11  |-  ( ( y  =  w  /\  x  =  z )  ->  ( y  C_  x  <->  w 
C_  z ) )
7271ancoms 454 . . . . . . . . . 10  |-  ( ( x  =  z  /\  y  =  w )  ->  ( y  C_  x  <->  w 
C_  z ) )
7370, 72anbi12d 715 . . . . . . . . 9  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( x  C_  B  /\  y  C_  x
)  <->  ( z  C_  B  /\  w  C_  z
) ) )
74 fveq2 5825 . . . . . . . . . 10  |-  ( y  =  w  ->  ( F `  y )  =  ( F `  w ) )
75 sseq12 3430 . . . . . . . . . 10  |-  ( ( ( F `  y
)  =  ( F `
 w )  /\  ( F `  x )  =  ( F `  z ) )  -> 
( ( F `  y )  C_  ( F `  x )  <->  ( F `  w ) 
C_  ( F `  z ) ) )
7674, 50, 75syl2anr 480 . . . . . . . . 9  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( F `  y )  C_  ( F `  x )  <->  ( F `  w ) 
C_  ( F `  z ) ) )
7773, 76imbi12d 321 . . . . . . . 8  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( ( x 
C_  B  /\  y  C_  x )  ->  ( F `  y )  C_  ( F `  x
) )  <->  ( (
z  C_  B  /\  w  C_  z )  -> 
( F `  w
)  C_  ( F `  z ) ) ) )
7877spc2gv 3112 . . . . . . 7  |-  ( ( z  e.  _V  /\  w  e.  _V )  ->  ( A. x A. y ( ( x 
C_  B  /\  y  C_  x )  ->  ( F `  y )  C_  ( F `  x
) )  ->  (
( z  C_  B  /\  w  C_  z )  ->  ( F `  w )  C_  ( F `  z )
) ) )
7943, 69, 78mp2an 676 . . . . . 6  |-  ( A. x A. y ( ( x  C_  B  /\  y  C_  x )  -> 
( F `  y
)  C_  ( F `  x ) )  -> 
( ( z  C_  B  /\  w  C_  z
)  ->  ( F `  w )  C_  ( F `  z )
) )
8068, 79syl 17 . . . . 5  |-  ( ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y
( ( x  C_  B  /\  y  C_  x
)  ->  ( x  C_  ( F `  x
)  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) )  ->  (
( z  C_  B  /\  w  C_  z )  ->  ( F `  w )  C_  ( F `  z )
) )
81803impib 1203 . . . 4  |-  ( ( ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y ( ( x 
C_  B  /\  y  C_  x )  ->  (
x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x
)  /\  ( F `  ( F `  x
) )  =  ( F `  x ) ) ) )  /\  z  C_  B  /\  w  C_  z )  ->  ( F `  w )  C_  ( F `  z
) )
8263simprd 464 . . . 4  |-  ( ( ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y ( ( x 
C_  B  /\  y  C_  x )  ->  (
x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x
)  /\  ( F `  ( F `  x
) )  =  ( F `  x ) ) ) )  /\  z  C_  B )  -> 
( F `  ( F `  z )
)  =  ( F `
 z ) )
8337, 38, 64, 81, 82ismrcd2 35453 . . 3  |-  ( ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y
( ( x  C_  B  /\  y  C_  x
)  ->  ( x  C_  ( F `  x
)  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) )  ->  F  =  (mrCls `  dom  ( F  i^i  _I  ) ) )
8437, 38, 64, 81, 82ismrcd1 35452 . . . 4  |-  ( ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y
( ( x  C_  B  /\  y  C_  x
)  ->  ( x  C_  ( F `  x
)  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) )  ->  dom  ( F  i^i  _I  )  e.  (Moore `  B )
)
85 fvssunirn 5848 . . . . . 6  |-  (Moore `  B )  C_  U. ran Moore
86 fndm 5636 . . . . . . 7  |-  (mrCls  Fn  U.
ran Moore  ->  dom mrCls  =  U. ran Moore )
871, 86ax-mp 5 . . . . . 6  |-  dom mrCls  =  U. ran Moore
8885, 87sseqtr4i 3440 . . . . 5  |-  (Moore `  B )  C_  dom mrCls
89 funfvima2 6100 . . . . 5  |-  ( ( Fun mrCls  /\  (Moore `  B
)  C_  dom mrCls )  -> 
( dom  ( F  i^i  _I  )  e.  (Moore `  B )  ->  (mrCls ` 
dom  ( F  i^i  _I  ) )  e.  (mrCls " (Moore `  B )
) ) )
903, 88, 89mp2an 676 . . . 4  |-  ( dom  ( F  i^i  _I  )  e.  (Moore `  B
)  ->  (mrCls `  dom  ( F  i^i  _I  )
)  e.  (mrCls "
(Moore `  B )
) )
9184, 90syl 17 . . 3  |-  ( ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y
( ( x  C_  B  /\  y  C_  x
)  ->  ( x  C_  ( F `  x
)  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) )  ->  (mrCls ` 
dom  ( F  i^i  _I  ) )  e.  (mrCls " (Moore `  B )
) )
9283, 91eqeltrd 2506 . 2  |-  ( ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y
( ( x  C_  B  /\  y  C_  x
)  ->  ( x  C_  ( F `  x
)  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) )  ->  F  e.  (mrCls " (Moore `  B
) ) )
9336, 92impbii 190 1  |-  ( F  e.  (mrCls " (Moore `  B ) )  <->  ( B  e.  _V  /\  F : ~P B --> ~P B  /\  A. x A. y ( ( x  C_  B  /\  y  C_  x )  ->  ( x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x )  /\  ( F `  ( F `  x )
)  =  ( F `
 x ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437    e. wcel 1872   E.wrex 2715   _Vcvv 3022    i^i cin 3378    C_ wss 3379   ~Pcpw 3924   U.cuni 4162    _I cid 4706   dom cdm 4796   ran crn 4797   "cima 4799   Fun wfun 5538    Fn wfn 5539   -->wf 5540   ` cfv 5544  Moorecmre 15431  mrClscmrc 15432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-int 4199  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-fv 5552  df-mre 15435  df-mrc 15436
This theorem is referenced by: (None)
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