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Theorem mreacs 13810
Description: Algebraicity is a composible property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
mreacs  |-  ( X  e.  V  ->  (ACS `  X )  e.  (Moore `  ~P X ) )

Proof of Theorem mreacs
Dummy variables  a 
b  c  x  d  e  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5668 . . 3  |-  ( x  =  X  ->  (ACS `  x )  =  (ACS
`  X ) )
2 pweq 3745 . . . 4  |-  ( x  =  X  ->  ~P x  =  ~P X
)
32fveq2d 5672 . . 3  |-  ( x  =  X  ->  (Moore `  ~P x )  =  (Moore `  ~P X ) )
41, 3eleq12d 2455 . 2  |-  ( x  =  X  ->  (
(ACS `  x )  e.  (Moore `  ~P x
)  <->  (ACS `  X )  e.  (Moore `  ~P X ) ) )
5 acsmre 13804 . . . . . . . 8  |-  ( a  e.  (ACS `  x
)  ->  a  e.  (Moore `  x ) )
6 mresspw 13744 . . . . . . . 8  |-  ( a  e.  (Moore `  x
)  ->  a  C_  ~P x )
75, 6syl 16 . . . . . . 7  |-  ( a  e.  (ACS `  x
)  ->  a  C_  ~P x )
8 vex 2902 . . . . . . . 8  |-  a  e. 
_V
98elpw 3748 . . . . . . 7  |-  ( a  e.  ~P ~P x  <->  a 
C_  ~P x )
107, 9sylibr 204 . . . . . 6  |-  ( a  e.  (ACS `  x
)  ->  a  e.  ~P ~P x )
1110ssriv 3295 . . . . 5  |-  (ACS `  x )  C_  ~P ~P x
1211a1i 11 . . . 4  |-  (  T. 
->  (ACS `  x )  C_ 
~P ~P x )
13 vex 2902 . . . . . . . 8  |-  x  e. 
_V
14 mremre 13756 . . . . . . . 8  |-  ( x  e.  _V  ->  (Moore `  x )  e.  (Moore `  ~P x ) )
1513, 14mp1i 12 . . . . . . 7  |-  ( a 
C_  (ACS `  x
)  ->  (Moore `  x
)  e.  (Moore `  ~P x ) )
165ssriv 3295 . . . . . . . 8  |-  (ACS `  x )  C_  (Moore `  x )
17 sstr 3299 . . . . . . . 8  |-  ( ( a  C_  (ACS `  x
)  /\  (ACS `  x
)  C_  (Moore `  x
) )  ->  a  C_  (Moore `  x )
)
1816, 17mpan2 653 . . . . . . 7  |-  ( a 
C_  (ACS `  x
)  ->  a  C_  (Moore `  x ) )
19 mrerintcl 13749 . . . . . . 7  |-  ( ( (Moore `  x )  e.  (Moore `  ~P x
)  /\  a  C_  (Moore `  x ) )  ->  ( ~P x  i^i  |^| a )  e.  (Moore `  x )
)
2015, 18, 19syl2anc 643 . . . . . 6  |-  ( a 
C_  (ACS `  x
)  ->  ( ~P x  i^i  |^| a )  e.  (Moore `  x )
)
21 ssel2 3286 . . . . . . . . . . . . . . . 16  |-  ( ( a  C_  (ACS `  x
)  /\  d  e.  a )  ->  d  e.  (ACS `  x )
)
2221acsmred 13808 . . . . . . . . . . . . . . 15  |-  ( ( a  C_  (ACS `  x
)  /\  d  e.  a )  ->  d  e.  (Moore `  x )
)
23 eqid 2387 . . . . . . . . . . . . . . 15  |-  (mrCls `  d )  =  (mrCls `  d )
2422, 23mrcssvd 13775 . . . . . . . . . . . . . 14  |-  ( ( a  C_  (ACS `  x
)  /\  d  e.  a )  ->  (
(mrCls `  d ) `  c )  C_  x
)
2524ralrimiva 2732 . . . . . . . . . . . . 13  |-  ( a 
C_  (ACS `  x
)  ->  A. d  e.  a  ( (mrCls `  d ) `  c
)  C_  x )
2625adantr 452 . . . . . . . . . . . 12  |-  ( ( a  C_  (ACS `  x
)  /\  c  e.  ~P x )  ->  A. d  e.  a  ( (mrCls `  d ) `  c
)  C_  x )
27 iunss 4073 . . . . . . . . . . . 12  |-  ( U_ d  e.  a  (
(mrCls `  d ) `  c )  C_  x  <->  A. d  e.  a  ( (mrCls `  d ) `  c )  C_  x
)
2826, 27sylibr 204 . . . . . . . . . . 11  |-  ( ( a  C_  (ACS `  x
)  /\  c  e.  ~P x )  ->  U_ d  e.  a  ( (mrCls `  d ) `  c
)  C_  x )
2913elpw2 4305 . . . . . . . . . . 11  |-  ( U_ d  e.  a  (
(mrCls `  d ) `  c )  e.  ~P x 
<-> 
U_ d  e.  a  ( (mrCls `  d
) `  c )  C_  x )
3028, 29sylibr 204 . . . . . . . . . 10  |-  ( ( a  C_  (ACS `  x
)  /\  c  e.  ~P x )  ->  U_ d  e.  a  ( (mrCls `  d ) `  c
)  e.  ~P x
)
31 eqid 2387 . . . . . . . . . 10  |-  ( c  e.  ~P x  |->  U_ d  e.  a  (
(mrCls `  d ) `  c ) )  =  ( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d
) `  c )
)
3230, 31fmptd 5832 . . . . . . . . 9  |-  ( a 
C_  (ACS `  x
)  ->  ( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) ) : ~P x
--> ~P x )
33 fssxp 5542 . . . . . . . . 9  |-  ( ( c  e.  ~P x  |-> 
U_ d  e.  a  ( (mrCls `  d
) `  c )
) : ~P x --> ~P x  ->  ( c  e.  ~P x  |->  U_ d  e.  a  (
(mrCls `  d ) `  c ) )  C_  ( ~P x  X.  ~P x ) )
3432, 33syl 16 . . . . . . . 8  |-  ( a 
C_  (ACS `  x
)  ->  ( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) )  C_  ( ~P x  X.  ~P x
) )
3513pwex 4323 . . . . . . . . 9  |-  ~P x  e.  _V
3635, 35xpex 4930 . . . . . . . 8  |-  ( ~P x  X.  ~P x
)  e.  _V
37 ssexg 4290 . . . . . . . 8  |-  ( ( ( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d
) `  c )
)  C_  ( ~P x  X.  ~P x )  /\  ( ~P x  X.  ~P x )  e. 
_V )  ->  (
c  e.  ~P x  |-> 
U_ d  e.  a  ( (mrCls `  d
) `  c )
)  e.  _V )
3834, 36, 37sylancl 644 . . . . . . 7  |-  ( a 
C_  (ACS `  x
)  ->  ( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) )  e.  _V )
3921adantlr 696 . . . . . . . . . . . . 13  |-  ( ( ( a  C_  (ACS `  x )  /\  b  e.  ~P x )  /\  d  e.  a )  ->  d  e.  (ACS `  x ) )
40 elpwi 3750 . . . . . . . . . . . . . 14  |-  ( b  e.  ~P x  -> 
b  C_  x )
4140ad2antlr 708 . . . . . . . . . . . . 13  |-  ( ( ( a  C_  (ACS `  x )  /\  b  e.  ~P x )  /\  d  e.  a )  ->  b  C_  x )
4223acsfiel2 13807 . . . . . . . . . . . . 13  |-  ( ( d  e.  (ACS `  x )  /\  b  C_  x )  ->  (
b  e.  d  <->  A. e  e.  ( ~P b  i^i 
Fin ) ( (mrCls `  d ) `  e
)  C_  b )
)
4339, 41, 42syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( a  C_  (ACS `  x )  /\  b  e.  ~P x )  /\  d  e.  a )  ->  ( b  e.  d  <->  A. e  e.  ( ~P b  i^i  Fin )
( (mrCls `  d
) `  e )  C_  b ) )
4443ralbidva 2665 . . . . . . . . . . 11  |-  ( ( a  C_  (ACS `  x
)  /\  b  e.  ~P x )  ->  ( A. d  e.  a 
b  e.  d  <->  A. d  e.  a  A. e  e.  ( ~P b  i^i 
Fin ) ( (mrCls `  d ) `  e
)  C_  b )
)
45 iunss 4073 . . . . . . . . . . . . 13  |-  ( U_ d  e.  a  (
(mrCls `  d ) `  e )  C_  b  <->  A. d  e.  a  ( (mrCls `  d ) `  e )  C_  b
)
4645ralbii 2673 . . . . . . . . . . . 12  |-  ( A. e  e.  ( ~P b  i^i  Fin ) U_ d  e.  a  (
(mrCls `  d ) `  e )  C_  b  <->  A. e  e.  ( ~P b  i^i  Fin ) A. d  e.  a 
( (mrCls `  d
) `  e )  C_  b )
47 ralcom 2811 . . . . . . . . . . . 12  |-  ( A. e  e.  ( ~P b  i^i  Fin ) A. d  e.  a  (
(mrCls `  d ) `  e )  C_  b  <->  A. d  e.  a  A. e  e.  ( ~P b  i^i  Fin ) ( (mrCls `  d ) `  e )  C_  b
)
4846, 47bitri 241 . . . . . . . . . . 11  |-  ( A. e  e.  ( ~P b  i^i  Fin ) U_ d  e.  a  (
(mrCls `  d ) `  e )  C_  b  <->  A. d  e.  a  A. e  e.  ( ~P b  i^i  Fin ) ( (mrCls `  d ) `  e )  C_  b
)
4944, 48syl6bbr 255 . . . . . . . . . 10  |-  ( ( a  C_  (ACS `  x
)  /\  b  e.  ~P x )  ->  ( A. d  e.  a 
b  e.  d  <->  A. e  e.  ( ~P b  i^i 
Fin ) U_ d  e.  a  ( (mrCls `  d ) `  e
)  C_  b )
)
50 elrint2 4034 . . . . . . . . . . 11  |-  ( b  e.  ~P x  -> 
( b  e.  ( ~P x  i^i  |^| a )  <->  A. d  e.  a  b  e.  d ) )
5150adantl 453 . . . . . . . . . 10  |-  ( ( a  C_  (ACS `  x
)  /\  b  e.  ~P x )  ->  (
b  e.  ( ~P x  i^i  |^| a
)  <->  A. d  e.  a  b  e.  d ) )
52 funmpt 5429 . . . . . . . . . . . . 13  |-  Fun  (
c  e.  ~P x  |-> 
U_ d  e.  a  ( (mrCls `  d
) `  c )
)
53 funiunfv 5934 . . . . . . . . . . . . 13  |-  ( Fun  ( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d
) `  c )
)  ->  U_ e  e.  ( ~P b  i^i 
Fin ) ( ( c  e.  ~P x  |-> 
U_ d  e.  a  ( (mrCls `  d
) `  c )
) `  e )  =  U. ( ( c  e.  ~P x  |->  U_ d  e.  a  (
(mrCls `  d ) `  c ) ) "
( ~P b  i^i 
Fin ) ) )
5452, 53ax-mp 8 . . . . . . . . . . . 12  |-  U_ e  e.  ( ~P b  i^i 
Fin ) ( ( c  e.  ~P x  |-> 
U_ d  e.  a  ( (mrCls `  d
) `  c )
) `  e )  =  U. ( ( c  e.  ~P x  |->  U_ d  e.  a  (
(mrCls `  d ) `  c ) ) "
( ~P b  i^i 
Fin ) )
5554sseq1i 3315 . . . . . . . . . . 11  |-  ( U_ e  e.  ( ~P b  i^i  Fin ) ( ( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d
) `  c )
) `  e )  C_  b  <->  U. ( ( c  e.  ~P x  |->  U_ d  e.  a  (
(mrCls `  d ) `  c ) ) "
( ~P b  i^i 
Fin ) )  C_  b )
56 iunss 4073 . . . . . . . . . . . 12  |-  ( U_ e  e.  ( ~P b  i^i  Fin ) ( ( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d
) `  c )
) `  e )  C_  b  <->  A. e  e.  ( ~P b  i^i  Fin ) ( ( c  e.  ~P x  |->  U_ d  e.  a  (
(mrCls `  d ) `  c ) ) `  e )  C_  b
)
57 inss1 3504 . . . . . . . . . . . . . . . . 17  |-  ( ~P b  i^i  Fin )  C_ 
~P b
58 sspwb 4354 . . . . . . . . . . . . . . . . . . 19  |-  ( b 
C_  x  <->  ~P b  C_ 
~P x )
5940, 58sylib 189 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  ~P x  ->  ~P b  C_  ~P x
)
6059adantl 453 . . . . . . . . . . . . . . . . 17  |-  ( ( a  C_  (ACS `  x
)  /\  b  e.  ~P x )  ->  ~P b  C_  ~P x )
6157, 60syl5ss 3302 . . . . . . . . . . . . . . . 16  |-  ( ( a  C_  (ACS `  x
)  /\  b  e.  ~P x )  ->  ( ~P b  i^i  Fin )  C_ 
~P x )
6261sselda 3291 . . . . . . . . . . . . . . 15  |-  ( ( ( a  C_  (ACS `  x )  /\  b  e.  ~P x )  /\  e  e.  ( ~P b  i^i  Fin ) )  ->  e  e.  ~P x )
6322, 23mrcssvd 13775 . . . . . . . . . . . . . . . . . . 19  |-  ( ( a  C_  (ACS `  x
)  /\  d  e.  a )  ->  (
(mrCls `  d ) `  e )  C_  x
)
6463ralrimiva 2732 . . . . . . . . . . . . . . . . . 18  |-  ( a 
C_  (ACS `  x
)  ->  A. d  e.  a  ( (mrCls `  d ) `  e
)  C_  x )
6564ad2antrr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  C_  (ACS `  x )  /\  b  e.  ~P x )  /\  e  e.  ( ~P b  i^i  Fin ) )  ->  A. d  e.  a  ( (mrCls `  d
) `  e )  C_  x )
66 iunss 4073 . . . . . . . . . . . . . . . . 17  |-  ( U_ d  e.  a  (
(mrCls `  d ) `  e )  C_  x  <->  A. d  e.  a  ( (mrCls `  d ) `  e )  C_  x
)
6765, 66sylibr 204 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  C_  (ACS `  x )  /\  b  e.  ~P x )  /\  e  e.  ( ~P b  i^i  Fin ) )  ->  U_ d  e.  a  ( (mrCls `  d
) `  e )  C_  x )
68 ssexg 4290 . . . . . . . . . . . . . . . 16  |-  ( (
U_ d  e.  a  ( (mrCls `  d
) `  e )  C_  x  /\  x  e. 
_V )  ->  U_ d  e.  a  ( (mrCls `  d ) `  e
)  e.  _V )
6967, 13, 68sylancl 644 . . . . . . . . . . . . . . 15  |-  ( ( ( a  C_  (ACS `  x )  /\  b  e.  ~P x )  /\  e  e.  ( ~P b  i^i  Fin ) )  ->  U_ d  e.  a  ( (mrCls `  d
) `  e )  e.  _V )
70 fveq2 5668 . . . . . . . . . . . . . . . . 17  |-  ( c  =  e  ->  (
(mrCls `  d ) `  c )  =  ( (mrCls `  d ) `  e ) )
7170iuneq2d 4060 . . . . . . . . . . . . . . . 16  |-  ( c  =  e  ->  U_ d  e.  a  ( (mrCls `  d ) `  c
)  =  U_ d  e.  a  ( (mrCls `  d ) `  e
) )
7271, 31fvmptg 5743 . . . . . . . . . . . . . . 15  |-  ( ( e  e.  ~P x  /\  U_ d  e.  a  ( (mrCls `  d
) `  e )  e.  _V )  ->  (
( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d
) `  c )
) `  e )  =  U_ d  e.  a  ( (mrCls `  d
) `  e )
)
7362, 69, 72syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( a  C_  (ACS `  x )  /\  b  e.  ~P x )  /\  e  e.  ( ~P b  i^i  Fin ) )  ->  ( ( c  e.  ~P x  |->  U_ d  e.  a  (
(mrCls `  d ) `  c ) ) `  e )  =  U_ d  e.  a  (
(mrCls `  d ) `  e ) )
7473sseq1d 3318 . . . . . . . . . . . . 13  |-  ( ( ( a  C_  (ACS `  x )  /\  b  e.  ~P x )  /\  e  e.  ( ~P b  i^i  Fin ) )  ->  ( ( ( c  e.  ~P x  |-> 
U_ d  e.  a  ( (mrCls `  d
) `  c )
) `  e )  C_  b  <->  U_ d  e.  a  ( (mrCls `  d
) `  e )  C_  b ) )
7574ralbidva 2665 . . . . . . . . . . . 12  |-  ( ( a  C_  (ACS `  x
)  /\  b  e.  ~P x )  ->  ( A. e  e.  ( ~P b  i^i  Fin )
( ( c  e. 
~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) ) `  e
)  C_  b  <->  A. e  e.  ( ~P b  i^i 
Fin ) U_ d  e.  a  ( (mrCls `  d ) `  e
)  C_  b )
)
7656, 75syl5bb 249 . . . . . . . . . . 11  |-  ( ( a  C_  (ACS `  x
)  /\  b  e.  ~P x )  ->  ( U_ e  e.  ( ~P b  i^i  Fin )
( ( c  e. 
~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) ) `  e
)  C_  b  <->  A. e  e.  ( ~P b  i^i 
Fin ) U_ d  e.  a  ( (mrCls `  d ) `  e
)  C_  b )
)
7755, 76syl5bbr 251 . . . . . . . . . 10  |-  ( ( a  C_  (ACS `  x
)  /\  b  e.  ~P x )  ->  ( U. ( ( c  e. 
~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) ) " ( ~P b  i^i  Fin )
)  C_  b  <->  A. e  e.  ( ~P b  i^i 
Fin ) U_ d  e.  a  ( (mrCls `  d ) `  e
)  C_  b )
)
7849, 51, 773bitr4d 277 . . . . . . . . 9  |-  ( ( a  C_  (ACS `  x
)  /\  b  e.  ~P x )  ->  (
b  e.  ( ~P x  i^i  |^| a
)  <->  U. ( ( c  e.  ~P x  |->  U_ d  e.  a  (
(mrCls `  d ) `  c ) ) "
( ~P b  i^i 
Fin ) )  C_  b ) )
7978ralrimiva 2732 . . . . . . . 8  |-  ( a 
C_  (ACS `  x
)  ->  A. b  e.  ~P  x ( b  e.  ( ~P x  i^i  |^| a )  <->  U. (
( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d
) `  c )
) " ( ~P b  i^i  Fin )
)  C_  b )
)
8032, 79jca 519 . . . . . . 7  |-  ( a 
C_  (ACS `  x
)  ->  ( (
c  e.  ~P x  |-> 
U_ d  e.  a  ( (mrCls `  d
) `  c )
) : ~P x --> ~P x  /\  A. b  e.  ~P  x ( b  e.  ( ~P x  i^i  |^| a )  <->  U. (
( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d
) `  c )
) " ( ~P b  i^i  Fin )
)  C_  b )
) )
81 feq1 5516 . . . . . . . . 9  |-  ( f  =  ( c  e. 
~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) )  ->  (
f : ~P x --> ~P x  <->  ( c  e. 
~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) ) : ~P x
--> ~P x ) )
82 imaeq1 5138 . . . . . . . . . . . . 13  |-  ( f  =  ( c  e. 
~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) )  ->  (
f " ( ~P b  i^i  Fin )
)  =  ( ( c  e.  ~P x  |-> 
U_ d  e.  a  ( (mrCls `  d
) `  c )
) " ( ~P b  i^i  Fin )
) )
8382unieqd 3968 . . . . . . . . . . . 12  |-  ( f  =  ( c  e. 
~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) )  ->  U. (
f " ( ~P b  i^i  Fin )
)  =  U. (
( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d
) `  c )
) " ( ~P b  i^i  Fin )
) )
8483sseq1d 3318 . . . . . . . . . . 11  |-  ( f  =  ( c  e. 
~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) )  ->  ( U. ( f " ( ~P b  i^i  Fin )
)  C_  b  <->  U. (
( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d
) `  c )
) " ( ~P b  i^i  Fin )
)  C_  b )
)
8584bibi2d 310 . . . . . . . . . 10  |-  ( f  =  ( c  e. 
~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) )  ->  (
( b  e.  ( ~P x  i^i  |^| a )  <->  U. (
f " ( ~P b  i^i  Fin )
)  C_  b )  <->  ( b  e.  ( ~P x  i^i  |^| a
)  <->  U. ( ( c  e.  ~P x  |->  U_ d  e.  a  (
(mrCls `  d ) `  c ) ) "
( ~P b  i^i 
Fin ) )  C_  b ) ) )
8685ralbidv 2669 . . . . . . . . 9  |-  ( f  =  ( c  e. 
~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) )  ->  ( A. b  e.  ~P  x ( b  e.  ( ~P x  i^i  |^| a )  <->  U. (
f " ( ~P b  i^i  Fin )
)  C_  b )  <->  A. b  e.  ~P  x
( b  e.  ( ~P x  i^i  |^| a )  <->  U. (
( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d
) `  c )
) " ( ~P b  i^i  Fin )
)  C_  b )
) )
8781, 86anbi12d 692 . . . . . . . 8  |-  ( f  =  ( c  e. 
~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) )  ->  (
( f : ~P x
--> ~P x  /\  A. b  e.  ~P  x
( b  e.  ( ~P x  i^i  |^| a )  <->  U. (
f " ( ~P b  i^i  Fin )
)  C_  b )
)  <->  ( ( c  e.  ~P x  |->  U_ d  e.  a  (
(mrCls `  d ) `  c ) ) : ~P x --> ~P x  /\  A. b  e.  ~P  x ( b  e.  ( ~P x  i^i  |^| a )  <->  U. (
( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d
) `  c )
) " ( ~P b  i^i  Fin )
)  C_  b )
) ) )
8887spcegv 2980 . . . . . . 7  |-  ( ( c  e.  ~P x  |-> 
U_ d  e.  a  ( (mrCls `  d
) `  c )
)  e.  _V  ->  ( ( ( c  e. 
~P x  |->  U_ d  e.  a  ( (mrCls `  d ) `  c
) ) : ~P x
--> ~P x  /\  A. b  e.  ~P  x
( b  e.  ( ~P x  i^i  |^| a )  <->  U. (
( c  e.  ~P x  |->  U_ d  e.  a  ( (mrCls `  d
) `  c )
) " ( ~P b  i^i  Fin )
)  C_  b )
)  ->  E. f
( f : ~P x
--> ~P x  /\  A. b  e.  ~P  x
( b  e.  ( ~P x  i^i  |^| a )  <->  U. (
f " ( ~P b  i^i  Fin )
)  C_  b )
) ) )
8938, 80, 88sylc 58 . . . . . 6  |-  ( a 
C_  (ACS `  x
)  ->  E. f
( f : ~P x
--> ~P x  /\  A. b  e.  ~P  x
( b  e.  ( ~P x  i^i  |^| a )  <->  U. (
f " ( ~P b  i^i  Fin )
)  C_  b )
) )
90 isacs 13803 . . . . . 6  |-  ( ( ~P x  i^i  |^| a )  e.  (ACS
`  x )  <->  ( ( ~P x  i^i  |^| a
)  e.  (Moore `  x )  /\  E. f ( f : ~P x --> ~P x  /\  A. b  e.  ~P  x ( b  e.  ( ~P x  i^i  |^| a )  <->  U. (
f " ( ~P b  i^i  Fin )
)  C_  b )
) ) )
9120, 89, 90sylanbrc 646 . . . . 5  |-  ( a 
C_  (ACS `  x
)  ->  ( ~P x  i^i  |^| a )  e.  (ACS `  x )
)
9291adantl 453 . . . 4  |-  ( (  T.  /\  a  C_  (ACS `  x ) )  ->  ( ~P x  i^i  |^| a )  e.  (ACS `  x )
)
9312, 92ismred2 13755 . . 3  |-  (  T. 
->  (ACS `  x )  e.  (Moore `  ~P x
) )
9493trud 1329 . 2  |-  (ACS `  x )  e.  (Moore `  ~P x )
954, 94vtoclg 2954 1  |-  ( X  e.  V  ->  (ACS `  X )  e.  (Moore `  ~P X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    T. wtru 1322   E.wex 1547    = wceq 1649    e. wcel 1717   A.wral 2649   _Vcvv 2899    i^i cin 3262    C_ wss 3263   ~Pcpw 3742   U.cuni 3957   |^|cint 3992   U_ciun 4035    e. cmpt 4207    X. cxp 4816   "cima 4821   Fun wfun 5388   -->wf 5390   ` cfv 5394   Fincfn 7045  Moorecmre 13734  mrClscmrc 13735  ACScacs 13737
This theorem is referenced by:  acsfn1  13813  acsfn1c  13814  acsfn2  13815  submacs  14692  subgacs  14902  nsgacs  14903  lssacs  15970  acsfn1p  27176  subrgacs  27177  sdrgacs  27178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-mre 13738  df-mrc 13739  df-acs 13741
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