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Theorem opsrval 16232
Description: The value of the "ordered power series" function. This is the same as mPwSer psrval 16126, but with the addition of a well-order on  I we can turn a strict order on 
R into a strict order on the power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
opsrval.s  |-  S  =  ( I mPwSer  R )
opsrval.o  |-  O  =  ( ( I ordPwSer  R
) `  T )
opsrval.b  |-  B  =  ( Base `  S
)
opsrval.q  |-  .<  =  ( lt `  R )
opsrval.c  |-  C  =  ( T  <bag  I )
opsrval.d  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
opsrval.l  |-  .<_  =  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( (
x `  z )  .<  ( y `  z
)  /\  A. w  e.  D  ( w C z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }
opsrval.i  |-  ( ph  ->  I  e.  V )
opsrval.r  |-  ( ph  ->  R  e.  W )
opsrval.t  |-  ( ph  ->  T  C_  ( I  X.  I ) )
Assertion
Ref Expression
opsrval  |-  ( ph  ->  O  =  ( S sSet  <. ( le `  ndx ) ,  .<_  >. )
)
Distinct variable groups:    w, h, x, y, z, I    ph, w, x, y, z    w, D, z    w, T, x, y, z    w, R, x, y, z
Allowed substitution hints:    ph( h)    B( x, y, z, w, h)    C( x, y, z, w, h)    D( x, y, h)    R( h)    S( x, y, z, w, h)    .< ( x, y, z, w, h)    T( h)   
.<_ ( x, y, z, w, h)    O( x, y, z, w, h)    V( x, y, z, w, h)    W( x, y, z, w, h)

Proof of Theorem opsrval
Dummy variables  r 
i  p  s  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opsrval.o . 2  |-  O  =  ( ( I ordPwSer  R
) `  T )
2 opsrval.i . . . . 5  |-  ( ph  ->  I  e.  V )
3 elex 2809 . . . . 5  |-  ( I  e.  V  ->  I  e.  _V )
42, 3syl 15 . . . 4  |-  ( ph  ->  I  e.  _V )
5 opsrval.r . . . . 5  |-  ( ph  ->  R  e.  W )
6 elex 2809 . . . . 5  |-  ( R  e.  W  ->  R  e.  _V )
75, 6syl 15 . . . 4  |-  ( ph  ->  R  e.  _V )
8 xpexg 4816 . . . . . 6  |-  ( ( I  e.  V  /\  I  e.  V )  ->  ( I  X.  I
)  e.  _V )
92, 2, 8syl2anc 642 . . . . 5  |-  ( ph  ->  ( I  X.  I
)  e.  _V )
10 pwexg 4210 . . . . 5  |-  ( ( I  X.  I )  e.  _V  ->  ~P ( I  X.  I
)  e.  _V )
11 mptexg 5761 . . . . 5  |-  ( ~P ( I  X.  I
)  e.  _V  ->  ( r  e.  ~P (
I  X.  I ) 
|->  ( S sSet  <. ( le `  ndx ) ,  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) }
>. ) )  e.  _V )
129, 10, 113syl 18 . . . 4  |-  ( ph  ->  ( r  e.  ~P ( I  X.  I
)  |->  ( S sSet  <. ( le `  ndx ) ,  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) }
>. ) )  e.  _V )
13 simpl 443 . . . . . . . 8  |-  ( ( i  =  I  /\  s  =  R )  ->  i  =  I )
1413, 13xpeq12d 4730 . . . . . . 7  |-  ( ( i  =  I  /\  s  =  R )  ->  ( i  X.  i
)  =  ( I  X.  I ) )
1514pweqd 3643 . . . . . 6  |-  ( ( i  =  I  /\  s  =  R )  ->  ~P ( i  X.  i )  =  ~P ( I  X.  I
) )
16 ovex 5899 . . . . . . . 8  |-  ( i mPwSer 
s )  e.  _V
1716a1i 10 . . . . . . 7  |-  ( ( i  =  I  /\  s  =  R )  ->  ( i mPwSer  s )  e.  _V )
18 id 19 . . . . . . . . . 10  |-  ( p  =  ( i mPwSer  s
)  ->  p  =  ( i mPwSer  s )
)
19 oveq12 5883 . . . . . . . . . 10  |-  ( ( i  =  I  /\  s  =  R )  ->  ( i mPwSer  s )  =  ( I mPwSer  R
) )
2018, 19sylan9eqr 2350 . . . . . . . . 9  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  p  =  ( I mPwSer  R ) )
21 opsrval.s . . . . . . . . 9  |-  S  =  ( I mPwSer  R )
2220, 21syl6eqr 2346 . . . . . . . 8  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  p  =  S )
2322fveq2d 5545 . . . . . . . . . . . . 13  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  ( Base `  p )  =  (
Base `  S )
)
24 opsrval.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  S
)
2523, 24syl6eqr 2346 . . . . . . . . . . . 12  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  ( Base `  p )  =  B )
2625sseq2d 3219 . . . . . . . . . . 11  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  ( {
x ,  y } 
C_  ( Base `  p
)  <->  { x ,  y }  C_  B )
)
27 ovex 5899 . . . . . . . . . . . . . . 15  |-  ( NN0 
^m  i )  e. 
_V
2827rabex 4181 . . . . . . . . . . . . . 14  |-  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  e.  _V
2928a1i 10 . . . . . . . . . . . . 13  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  e.  _V )
3013adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  i  =  I )
3130oveq2d 5890 . . . . . . . . . . . . . . 15  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  ( NN0  ^m  i )  =  ( NN0  ^m  I ) )
32 rabeq 2795 . . . . . . . . . . . . . . 15  |-  ( ( NN0  ^m  i )  =  ( NN0  ^m  I )  ->  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } )
3331, 32syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } )
34 opsrval.d . . . . . . . . . . . . . 14  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
3533, 34syl6eqr 2346 . . . . . . . . . . . . 13  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  =  D )
36 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s
) )  /\  d  =  D )  ->  d  =  D )
37 simpllr 735 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s
) )  /\  d  =  D )  ->  s  =  R )
3837fveq2d 5545 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s
) )  /\  d  =  D )  ->  ( lt `  s )  =  ( lt `  R
) )
39 opsrval.q . . . . . . . . . . . . . . . . 17  |-  .<  =  ( lt `  R )
4038, 39syl6eqr 2346 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s
) )  /\  d  =  D )  ->  ( lt `  s )  = 
.<  )
4140breqd 4050 . . . . . . . . . . . . . . 15  |-  ( ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s
) )  /\  d  =  D )  ->  (
( x `  z
) ( lt `  s ) ( y `
 z )  <->  ( x `  z )  .<  (
y `  z )
) )
4230adantr 451 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s
) )  /\  d  =  D )  ->  i  =  I )
4342oveq2d 5890 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s
) )  /\  d  =  D )  ->  (
r  <bag  i )  =  ( r  <bag  I ) )
4443breqd 4050 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s
) )  /\  d  =  D )  ->  (
w ( r  <bag  i ) z  <->  w (
r  <bag  I ) z ) )
4544imbi1d 308 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s
) )  /\  d  =  D )  ->  (
( w ( r  <bag  i ) z  -> 
( x `  w
)  =  ( y `
 w ) )  <-> 
( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) ) )
4636, 45raleqbidv 2761 . . . . . . . . . . . . . . 15  |-  ( ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s
) )  /\  d  =  D )  ->  ( A. w  e.  d 
( w ( r  <bag  i ) z  -> 
( x `  w
)  =  ( y `
 w ) )  <->  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) ) )
4741, 46anbi12d 691 . . . . . . . . . . . . . 14  |-  ( ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s
) )  /\  d  =  D )  ->  (
( ( x `  z ) ( lt
`  s ) ( y `  z )  /\  A. w  e.  d  ( w ( r  <bag  i ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  <->  ( ( x `
 z )  .< 
( y `  z
)  /\  A. w  e.  D  ( w
( r  <bag  I ) z  ->  ( x `  w )  =  ( y `  w ) ) ) ) )
4836, 47rexeqbidv 2762 . . . . . . . . . . . . 13  |-  ( ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s
) )  /\  d  =  D )  ->  ( E. z  e.  d 
( ( x `  z ) ( lt
`  s ) ( y `  z )  /\  A. w  e.  d  ( w ( r  <bag  i ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  <->  E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) ) ) )
4929, 35, 48sbcied2 3041 . . . . . . . . . . . 12  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  ( [. { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  /  d ]. E. z  e.  d  (
( x `  z
) ( lt `  s ) ( y `
 z )  /\  A. w  e.  d  ( w ( r  <bag  i ) z  ->  (
x `  w )  =  ( y `  w ) ) )  <->  E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) ) ) )
5049orbi1d 683 . . . . . . . . . . 11  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  ( ( [. { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  /  d ]. E. z  e.  d  (
( x `  z
) ( lt `  s ) ( y `
 z )  /\  A. w  e.  d  ( w ( r  <bag  i ) z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y )  <->  ( E. z  e.  D  ( (
x `  z )  .<  ( y `  z
)  /\  A. w  e.  D  ( w
( r  <bag  I ) z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) )
5126, 50anbi12d 691 . . . . . . . . . 10  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  ( ( { x ,  y }  C_  ( Base `  p )  /\  ( [. { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  /  d ]. E. z  e.  d  (
( x `  z
) ( lt `  s ) ( y `
 z )  /\  A. w  e.  d  ( w ( r  <bag  i ) z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) )  <->  ( {
x ,  y } 
C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) ) )
5251opabbidv 4098 . . . . . . . . 9  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  p )  /\  ( [. { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  /  d ]. E. z  e.  d  (
( x `  z
) ( lt `  s ) ( y `
 z )  /\  A. w  e.  d  ( w ( r  <bag  i ) z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }  =  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) } )
5352opeq2d 3819 . . . . . . . 8  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  <. ( le
`  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  p )  /\  ( [. { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  / 
d ]. E. z  e.  d  ( ( x `
 z ) ( lt `  s ) ( y `  z
)  /\  A. w  e.  d  ( w
( r  <bag  i ) z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >.  =  <. ( le `  ndx ) ,  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) }
>. )
5422, 53oveq12d 5892 . . . . . . 7  |-  ( ( ( i  =  I  /\  s  =  R )  /\  p  =  ( i mPwSer  s ) )  ->  ( p sSet  <.
( le `  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  p )  /\  ( [. { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  /  d ]. E. z  e.  d  (
( x `  z
) ( lt `  s ) ( y `
 z )  /\  A. w  e.  d  ( w ( r  <bag  i ) z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >. )  =  ( S sSet  <. ( le `  ndx ) ,  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) }
>. ) )
5517, 54csbied 3136 . . . . . 6  |-  ( ( i  =  I  /\  s  =  R )  ->  [_ ( i mPwSer  s
)  /  p ]_ ( p sSet  <. ( le
`  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  p )  /\  ( [. { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  / 
d ]. E. z  e.  d  ( ( x `
 z ) ( lt `  s ) ( y `  z
)  /\  A. w  e.  d  ( w
( r  <bag  i ) z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >. )  =  ( S sSet  <. ( le `  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) }
>. ) )
5615, 55mpteq12dv 4114 . . . . 5  |-  ( ( i  =  I  /\  s  =  R )  ->  ( r  e.  ~P ( i  X.  i
)  |->  [_ ( i mPwSer  s
)  /  p ]_ ( p sSet  <. ( le
`  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  p )  /\  ( [. { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  / 
d ]. E. z  e.  d  ( ( x `
 z ) ( lt `  s ) ( y `  z
)  /\  A. w  e.  d  ( w
( r  <bag  i ) z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >. ) )  =  ( r  e.  ~P ( I  X.  I
)  |->  ( S sSet  <. ( le `  ndx ) ,  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) }
>. ) ) )
57 df-opsr 16122 . . . . 5  |- ordPwSer  =  ( i  e.  _V , 
s  e.  _V  |->  ( r  e.  ~P (
i  X.  i ) 
|->  [_ ( i mPwSer  s
)  /  p ]_ ( p sSet  <. ( le
`  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  p )  /\  ( [. { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  / 
d ]. E. z  e.  d  ( ( x `
 z ) ( lt `  s ) ( y `  z
)  /\  A. w  e.  d  ( w
( r  <bag  i ) z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >. ) ) )
5856, 57ovmpt2ga 5993 . . . 4  |-  ( ( I  e.  _V  /\  R  e.  _V  /\  (
r  e.  ~P (
I  X.  I ) 
|->  ( S sSet  <. ( le `  ndx ) ,  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) }
>. ) )  e.  _V )  ->  ( I ordPwSer  R
)  =  ( r  e.  ~P ( I  X.  I )  |->  ( S sSet  <. ( le `  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) }
>. ) ) )
594, 7, 12, 58syl3anc 1182 . . 3  |-  ( ph  ->  ( I ordPwSer  R )  =  ( r  e. 
~P ( I  X.  I )  |->  ( S sSet  <. ( le `  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) }
>. ) ) )
60 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  r  =  T )  ->  r  =  T )
6160oveq1d 5889 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  r  =  T )  ->  (
r  <bag  I )  =  ( T  <bag  I ) )
62 opsrval.c . . . . . . . . . . . . . . 15  |-  C  =  ( T  <bag  I )
6361, 62syl6eqr 2346 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  r  =  T )  ->  (
r  <bag  I )  =  C )
6463breqd 4050 . . . . . . . . . . . . 13  |-  ( (
ph  /\  r  =  T )  ->  (
w ( r  <bag  I ) z  <->  w C
z ) )
6564imbi1d 308 . . . . . . . . . . . 12  |-  ( (
ph  /\  r  =  T )  ->  (
( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) )  <-> 
( w C z  ->  ( x `  w )  =  ( y `  w ) ) ) )
6665ralbidv 2576 . . . . . . . . . . 11  |-  ( (
ph  /\  r  =  T )  ->  ( A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) )  <->  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) ) )
6766anbi2d 684 . . . . . . . . . 10  |-  ( (
ph  /\  r  =  T )  ->  (
( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  <->  ( ( x `
 z )  .< 
( y `  z
)  /\  A. w  e.  D  ( w C z  ->  (
x `  w )  =  ( y `  w ) ) ) ) )
6867rexbidv 2577 . . . . . . . . 9  |-  ( (
ph  /\  r  =  T )  ->  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  <->  E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) ) ) )
6968orbi1d 683 . . . . . . . 8  |-  ( (
ph  /\  r  =  T )  ->  (
( E. z  e.  D  ( ( x `
 z )  .< 
( y `  z
)  /\  A. w  e.  D  ( w
( r  <bag  I ) z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y )  <->  ( E. z  e.  D  (
( x `  z
)  .<  ( y `  z )  /\  A. w  e.  D  (
w C z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) )
7069anbi2d 684 . . . . . . 7  |-  ( (
ph  /\  r  =  T )  ->  (
( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `
 z )  .< 
( y `  z
)  /\  A. w  e.  D  ( w
( r  <bag  I ) z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) )  <-> 
( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `
 z )  .< 
( y `  z
)  /\  A. w  e.  D  ( w C z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) ) )
7170opabbidv 4098 . . . . . 6  |-  ( (
ph  /\  r  =  T )  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) }  =  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } )
72 opsrval.l . . . . . 6  |-  .<_  =  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( (
x `  z )  .<  ( y `  z
)  /\  A. w  e.  D  ( w C z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }
7371, 72syl6eqr 2346 . . . . 5  |-  ( (
ph  /\  r  =  T )  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) }  =  .<_  )
7473opeq2d 3819 . . . 4  |-  ( (
ph  /\  r  =  T )  ->  <. ( le `  ndx ) ,  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) }
>.  =  <. ( le
`  ndx ) ,  .<_  >.
)
7574oveq2d 5890 . . 3  |-  ( (
ph  /\  r  =  T )  ->  ( S sSet  <. ( le `  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w ( r  <bag  I ) z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) }
>. )  =  ( S sSet  <. ( le `  ndx ) ,  .<_  >. )
)
76 opsrval.t . . . 4  |-  ( ph  ->  T  C_  ( I  X.  I ) )
77 elpw2g 4190 . . . . 5  |-  ( ( I  X.  I )  e.  _V  ->  ( T  e.  ~P (
I  X.  I )  <-> 
T  C_  ( I  X.  I ) ) )
789, 77syl 15 . . . 4  |-  ( ph  ->  ( T  e.  ~P ( I  X.  I
)  <->  T  C_  ( I  X.  I ) ) )
7976, 78mpbird 223 . . 3  |-  ( ph  ->  T  e.  ~P (
I  X.  I ) )
80 ovex 5899 . . . 4  |-  ( S sSet  <. ( le `  ndx ) ,  .<_  >. )  e.  _V
8180a1i 10 . . 3  |-  ( ph  ->  ( S sSet  <. ( le `  ndx ) , 
.<_  >. )  e.  _V )
8259, 75, 79, 81fvmptd 5622 . 2  |-  ( ph  ->  ( ( I ordPwSer  R
) `  T )  =  ( S sSet  <. ( le `  ndx ) ,  .<_  >. ) )
831, 82syl5eq 2340 1  |-  ( ph  ->  O  =  ( S sSet  <. ( le `  ndx ) ,  .<_  >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801   [.wsbc 3004   [_csb 3094    C_ wss 3165   ~Pcpw 3638   {cpr 3654   <.cop 3656   class class class wbr 4039   {copab 4092    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Fincfn 6879   NNcn 9762   NN0cn0 9981   ndxcnx 13161   sSet csts 13162   Basecbs 13164   lecple 13231   ltcplt 14091   mPwSer cmps 16103    <bag cltb 16110   ordPwSer copws 16111
This theorem is referenced by:  opsrle  16233  opsrval2  16234
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-opsr 16122
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