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Theorem ismrc 26776
Description: A function is a Moore closure operator iff it satisfies mrcssid 13519, mrcss 13518, and mrcidm 13521. (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 13509 . . . . 5  |- mrCls  Fn  U. ran Moore
2 fnfun 5341 . . . . 5  |-  (mrCls  Fn  U.
ran Moore  ->  Fun mrCls )
31, 2ax-mp 8 . . . 4  |-  Fun mrCls
4 fvelima 5574 . . . 4  |-  ( ( Fun mrCls  /\  F  e.  (mrCls " (Moore `  B )
) )  ->  E. z  e.  (Moore `  B )
(mrCls `  z )  =  F )
53, 4mpan 651 . . 3  |-  ( F  e.  (mrCls " (Moore `  B ) )  ->  E. z  e.  (Moore `  B ) (mrCls `  z )  =  F )
6 elfvex 5555 . . . . . 6  |-  ( z  e.  (Moore `  B
)  ->  B  e.  _V )
7 eqid 2283 . . . . . . . 8  |-  (mrCls `  z )  =  (mrCls `  z )
87mrcf 13511 . . . . . . 7  |-  ( z  e.  (Moore `  B
)  ->  (mrCls `  z
) : ~P B --> z )
9 mresspw 13494 . . . . . . 7  |-  ( z  e.  (Moore `  B
)  ->  z  C_  ~P B )
10 fss 5397 . . . . . . 7  |-  ( ( (mrCls `  z ) : ~P B --> z  /\  z  C_  ~P B )  ->  (mrCls `  z
) : ~P B --> ~P B )
118, 9, 10syl2anc 642 . . . . . 6  |-  ( z  e.  (Moore `  B
)  ->  (mrCls `  z
) : ~P B --> ~P B )
127mrcssid 13519 . . . . . . . . . 10  |-  ( ( z  e.  (Moore `  B )  /\  x  C_  B )  ->  x  C_  ( (mrCls `  z
) `  x )
)
1312adantrr 697 . . . . . . . . 9  |-  ( ( z  e.  (Moore `  B )  /\  (
x  C_  B  /\  y  C_  x ) )  ->  x  C_  (
(mrCls `  z ) `  x ) )
147mrcss 13518 . . . . . . . . . . 11  |-  ( ( z  e.  (Moore `  B )  /\  y  C_  x  /\  x  C_  B )  ->  (
(mrCls `  z ) `  y )  C_  (
(mrCls `  z ) `  x ) )
15143expb 1152 . . . . . . . . . 10  |-  ( ( z  e.  (Moore `  B )  /\  (
y  C_  x  /\  x  C_  B ) )  ->  ( (mrCls `  z ) `  y
)  C_  ( (mrCls `  z ) `  x
) )
1615ancom2s 777 . . . . . . . . 9  |-  ( ( z  e.  (Moore `  B )  /\  (
x  C_  B  /\  y  C_  x ) )  ->  ( (mrCls `  z ) `  y
)  C_  ( (mrCls `  z ) `  x
) )
177mrcidm 13521 . . . . . . . . . 10  |-  ( ( z  e.  (Moore `  B )  /\  x  C_  B )  ->  (
(mrCls `  z ) `  ( (mrCls `  z
) `  x )
)  =  ( (mrCls `  z ) `  x
) )
1817adantrr 697 . . . . . . . . 9  |-  ( ( z  e.  (Moore `  B )  /\  (
x  C_  B  /\  y  C_  x ) )  ->  ( (mrCls `  z ) `  (
(mrCls `  z ) `  x ) )  =  ( (mrCls `  z
) `  x )
)
1913, 16, 183jca 1132 . . . . . . . 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
) ) )
2019ex 423 . . . . . . 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
) ) ) )
2120alrimivv 1618 . . . . . 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 )
) ) )
226, 11, 213jca 1132 . . . . 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
) ) ) ) )
23 feq1 5375 . . . . . 6  |-  ( (mrCls `  z )  =  F  ->  ( (mrCls `  z ) : ~P B
--> ~P B  <->  F : ~P B --> ~P B ) )
24 fveq1 5524 . . . . . . . . . 10  |-  ( (mrCls `  z )  =  F  ->  ( (mrCls `  z ) `  x
)  =  ( F `
 x ) )
2524sseq2d 3206 . . . . . . . . 9  |-  ( (mrCls `  z )  =  F  ->  ( x  C_  ( (mrCls `  z ) `  x )  <->  x  C_  ( F `  x )
) )
26 fveq1 5524 . . . . . . . . . 10  |-  ( (mrCls `  z )  =  F  ->  ( (mrCls `  z ) `  y
)  =  ( F `
 y ) )
2726, 24sseq12d 3207 . . . . . . . . 9  |-  ( (mrCls `  z )  =  F  ->  ( ( (mrCls `  z ) `  y
)  C_  ( (mrCls `  z ) `  x
)  <->  ( F `  y )  C_  ( F `  x )
) )
28 id 19 . . . . . . . . . . 11  |-  ( (mrCls `  z )  =  F  ->  (mrCls `  z
)  =  F )
2928, 24fveq12d 5531 . . . . . . . . . 10  |-  ( (mrCls `  z )  =  F  ->  ( (mrCls `  z ) `  (
(mrCls `  z ) `  x ) )  =  ( F `  ( F `  x )
) )
3029, 24eqeq12d 2297 . . . . . . . . 9  |-  ( (mrCls `  z )  =  F  ->  ( ( (mrCls `  z ) `  (
(mrCls `  z ) `  x ) )  =  ( (mrCls `  z
) `  x )  <->  ( F `  ( F `
 x ) )  =  ( F `  x ) ) )
3125, 27, 303anbi123d 1252 . . . . . . . 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 ) ) ) )
3231imbi2d 307 . . . . . . 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
) ) ) ) )
33322albidv 1613 . . . . . 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 ) ) ) ) )
3423, 333anbi23d 1255 . . . . 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 ) ) ) ) ) )
3522, 34syl5ibcom 211 . . . 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 ) ) ) ) ) )
3635rexlimiv 2661 . . 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 ) ) ) ) )
375, 36syl 15 . 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 ) ) ) ) )
38 simp1 955 . . . 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 )
39 simp2 956 . . . 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
)
40 ssid 3197 . . . . . . 7  |-  z  C_  z
41 3simpb 953 . . . . . . . . . . 11  |-  ( ( x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x
)  /\  ( F `  ( F `  x
) )  =  ( F `  x ) )  ->  ( x  C_  ( F `  x
)  /\  ( F `  ( F `  x
) )  =  ( F `  x ) ) )
4241imim2i 13 . . . . . . . . . 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 ) ) ) )
43422alimi 1547 . . . . . . . . 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 ) ) ) )
44 vex 2791 . . . . . . . . . 10  |-  z  e. 
_V
45 sseq1 3199 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
x  C_  B  <->  z  C_  B ) )
4645adantr 451 . . . . . . . . . . . . 13  |-  ( ( x  =  z  /\  y  =  z )  ->  ( x  C_  B  <->  z 
C_  B ) )
47 sseq12 3201 . . . . . . . . . . . . . 14  |-  ( ( y  =  z  /\  x  =  z )  ->  ( y  C_  x  <->  z 
C_  z ) )
4847ancoms 439 . . . . . . . . . . . . 13  |-  ( ( x  =  z  /\  y  =  z )  ->  ( y  C_  x  <->  z 
C_  z ) )
4946, 48anbi12d 691 . . . . . . . . . . . 12  |-  ( ( x  =  z  /\  y  =  z )  ->  ( ( x  C_  B  /\  y  C_  x
)  <->  ( z  C_  B  /\  z  C_  z
) ) )
50 id 19 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  x  =  z )
51 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
5250, 51sseq12d 3207 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
x  C_  ( F `  x )  <->  z  C_  ( F `  z ) ) )
5352adantr 451 . . . . . . . . . . . . 13  |-  ( ( x  =  z  /\  y  =  z )  ->  ( x  C_  ( F `  x )  <->  z 
C_  ( F `  z ) ) )
5451fveq2d 5529 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  z )
) )
5554, 51eqeq12d 2297 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
( F `  ( F `  x )
)  =  ( F `
 x )  <->  ( F `  ( F `  z
) )  =  ( F `  z ) ) )
5655adantr 451 . . . . . . . . . . . . 13  |-  ( ( x  =  z  /\  y  =  z )  ->  ( ( F `  ( F `  x ) )  =  ( F `
 x )  <->  ( F `  ( F `  z
) )  =  ( F `  z ) ) )
5753, 56anbi12d 691 . . . . . . . . . . . 12  |-  ( ( x  =  z  /\  y  =  z )  ->  ( ( x  C_  ( F `  x )  /\  ( F `  ( F `  x ) )  =  ( F `
 x ) )  <-> 
( z  C_  ( F `  z )  /\  ( F `  ( F `  z )
)  =  ( F `
 z ) ) ) )
5849, 57imbi12d 311 . . . . . . . . . . 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 ) ) ) ) )
5958spc2gv 2871 . . . . . . . . . 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 ) ) ) ) )
6044, 44, 59mp2an 653 . . . . . . . . 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 ) ) ) )
6143, 60syl 15 . . . . . . . 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 ) ) ) )
62613ad2ant3 978 . . . . . . 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 ) ) ) )
6340, 62mpan2i 658 . . . . . 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
) ) ) )
6463imp 418 . . . . 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 ) ) )
6564simpld 445 . . . 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 ) )
66 simp2 956 . . . . . . . . 9  |-  ( ( x  C_  ( F `  x )  /\  ( F `  y )  C_  ( F `  x
)  /\  ( F `  ( F `  x
) )  =  ( F `  x ) )  ->  ( F `  y )  C_  ( F `  x )
)
6766imim2i 13 . . . . . . . 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 ) ) )
68672alimi 1547 . . . . . . 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 )
) )
69683ad2ant3 978 . . . . . 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
) ) )
70 vex 2791 . . . . . . 7  |-  w  e. 
_V
7145adantr 451 . . . . . . . . . 10  |-  ( ( x  =  z  /\  y  =  w )  ->  ( x  C_  B  <->  z 
C_  B ) )
72 sseq12 3201 . . . . . . . . . . 11  |-  ( ( y  =  w  /\  x  =  z )  ->  ( y  C_  x  <->  w 
C_  z ) )
7372ancoms 439 . . . . . . . . . 10  |-  ( ( x  =  z  /\  y  =  w )  ->  ( y  C_  x  <->  w 
C_  z ) )
7471, 73anbi12d 691 . . . . . . . . 9  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( x  C_  B  /\  y  C_  x
)  <->  ( z  C_  B  /\  w  C_  z
) ) )
75 fveq2 5525 . . . . . . . . . 10  |-  ( y  =  w  ->  ( F `  y )  =  ( F `  w ) )
76 sseq12 3201 . . . . . . . . . 10  |-  ( ( ( F `  y
)  =  ( F `
 w )  /\  ( F `  x )  =  ( F `  z ) )  -> 
( ( F `  y )  C_  ( F `  x )  <->  ( F `  w ) 
C_  ( F `  z ) ) )
7775, 51, 76syl2anr 464 . . . . . . . . 9  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( F `  y )  C_  ( F `  x )  <->  ( F `  w ) 
C_  ( F `  z ) ) )
7874, 77imbi12d 311 . . . . . . . 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 ) ) ) )
7978spc2gv 2871 . . . . . . 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 )
) ) )
8044, 70, 79mp2an 653 . . . . . 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 )
) )
8169, 80syl 15 . . . . 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 )
) )
82813impib 1149 . . . 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
) )
8364simprd 449 . . . 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 ) )
8438, 39, 65, 82, 83ismrcd2 26774 . . 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  ) ) )
8538, 39, 65, 82, 83ismrcd1 26773 . . . 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 )
)
86 fvssunirn 5551 . . . . . 6  |-  (Moore `  B )  C_  U. ran Moore
87 fndm 5343 . . . . . . 7  |-  (mrCls  Fn  U.
ran Moore  ->  dom mrCls  =  U. ran Moore )
881, 87ax-mp 8 . . . . . 6  |-  dom mrCls  =  U. ran Moore
8986, 88sseqtr4i 3211 . . . . 5  |-  (Moore `  B )  C_  dom mrCls
90 funfvima2 5754 . . . . 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 )
) ) )
913, 89, 90mp2an 653 . . . 4  |-  ( dom  ( F  i^i  _I  )  e.  (Moore `  B
)  ->  (mrCls `  dom  ( F  i^i  _I  )
)  e.  (mrCls "
(Moore `  B )
) )
9285, 91syl 15 . . 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 )
) )
9384, 92eqeltrd 2357 . 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
) ) )
9437, 93impbii 180 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827    _I cid 4304   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  Moorecmre 13484  mrClscmrc 13485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-mre 13488  df-mrc 13489
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