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Theorem msrfval 28875
Description: Value of the reduct of a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msrfval.v  |-  V  =  (mVars `  T )
msrfval.p  |-  P  =  (mPreSt `  T )
msrfval.r  |-  R  =  (mStRed `  T )
Assertion
Ref Expression
msrfval  |-  R  =  ( s  e.  P  |-> 
[_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  / 
a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
Distinct variable groups:    h, a,
s, z, P    T, a, h, s    z, V
Allowed substitution hints:    R( z, h, s, a)    T( z)    V( h, s, a)

Proof of Theorem msrfval
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 msrfval.r . 2  |-  R  =  (mStRed `  T )
2 fveq2 5856 . . . . . 6  |-  ( t  =  T  ->  (mPreSt `  t )  =  (mPreSt `  T ) )
3 msrfval.p . . . . . 6  |-  P  =  (mPreSt `  T )
42, 3syl6eqr 2502 . . . . 5  |-  ( t  =  T  ->  (mPreSt `  t )  =  P )
5 fveq2 5856 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  (mVars `  t )  =  (mVars `  T ) )
6 msrfval.v . . . . . . . . . . . . 13  |-  V  =  (mVars `  T )
75, 6syl6eqr 2502 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (mVars `  t )  =  V )
87imaeq1d 5326 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
(mVars `  t ) " ( h  u. 
{ a } ) )  =  ( V
" ( h  u. 
{ a } ) ) )
98unieqd 4244 . . . . . . . . . 10  |-  ( t  =  T  ->  U. (
(mVars `  t ) " ( h  u. 
{ a } ) )  =  U. ( V " ( h  u. 
{ a } ) ) )
109csbeq1d 3427 . . . . . . . . 9  |-  ( t  =  T  ->  [_ U. ( (mVars `  t ) " ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
)  =  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) )
1110ineq2d 3685 . . . . . . . 8  |-  ( t  =  T  ->  (
( 1st `  ( 1st `  s ) )  i^i  [_ U. ( (mVars `  t ) " (
h  u.  { a } ) )  / 
z ]_ ( z  X.  z ) )  =  ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( V
" ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) )
1211oteq1d 4214 . . . . . . 7  |-  ( t  =  T  ->  <. (
( 1st `  ( 1st `  s ) )  i^i  [_ U. ( (mVars `  t ) " (
h  u.  { a } ) )  / 
z ]_ ( z  X.  z ) ) ,  h ,  a >.  =  <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
1312csbeq2dv 3821 . . . . . 6  |-  ( t  =  T  ->  [_ ( 2nd `  s )  / 
a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( (mVars `  t ) " ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >.  =  [_ ( 2nd `  s )  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( V
" ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
1413csbeq2dv 3821 . . . . 5  |-  ( t  =  T  ->  [_ ( 2nd `  ( 1st `  s
) )  /  h ]_ [_ ( 2nd `  s
)  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( (mVars `  t ) " (
h  u.  { a } ) )  / 
z ]_ ( z  X.  z ) ) ,  h ,  a >.  =  [_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  / 
a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
154, 14mpteq12dv 4515 . . . 4  |-  ( t  =  T  ->  (
s  e.  (mPreSt `  t )  |->  [_ ( 2nd `  ( 1st `  s
) )  /  h ]_ [_ ( 2nd `  s
)  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( (mVars `  t ) " (
h  u.  { a } ) )  / 
z ]_ ( z  X.  z ) ) ,  h ,  a >.
)  =  ( s  e.  P  |->  [_ ( 2nd `  ( 1st `  s
) )  /  h ]_ [_ ( 2nd `  s
)  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( V
" ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
)
16 df-msr 28832 . . . 4  |- mStRed  =  ( t  e.  _V  |->  ( s  e.  (mPreSt `  t )  |->  [_ ( 2nd `  ( 1st `  s
) )  /  h ]_ [_ ( 2nd `  s
)  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( (mVars `  t ) " (
h  u.  { a } ) )  / 
z ]_ ( z  X.  z ) ) ,  h ,  a >.
) )
17 fvex 5866 . . . . . 6  |-  (mPreSt `  T )  e.  _V
183, 17eqeltri 2527 . . . . 5  |-  P  e. 
_V
1918mptex 6128 . . . 4  |-  ( s  e.  P  |->  [_ ( 2nd `  ( 1st `  s
) )  /  h ]_ [_ ( 2nd `  s
)  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( V
" ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )  e.  _V
2015, 16, 19fvmpt 5941 . . 3  |-  ( T  e.  _V  ->  (mStRed `  T )  =  ( s  e.  P  |->  [_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  /  a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
)
21 mpt0 5698 . . . . 5  |-  ( s  e.  (/)  |->  [_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  / 
a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )  =  (/)
2221eqcomi 2456 . . . 4  |-  (/)  =  ( s  e.  (/)  |->  [_ ( 2nd `  ( 1st `  s
) )  /  h ]_ [_ ( 2nd `  s
)  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( V
" ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
23 fvprc 5850 . . . 4  |-  ( -.  T  e.  _V  ->  (mStRed `  T )  =  (/) )
24 fvprc 5850 . . . . . 6  |-  ( -.  T  e.  _V  ->  (mPreSt `  T )  =  (/) )
253, 24syl5eq 2496 . . . . 5  |-  ( -.  T  e.  _V  ->  P  =  (/) )
2625mpteq1d 4518 . . . 4  |-  ( -.  T  e.  _V  ->  ( s  e.  P  |->  [_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  /  a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )  =  ( s  e.  (/)  |->  [_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  / 
a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
)
2722, 23, 263eqtr4a 2510 . . 3  |-  ( -.  T  e.  _V  ->  (mStRed `  T )  =  ( s  e.  P  |->  [_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  /  a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
)
2820, 27pm2.61i 164 . 2  |-  (mStRed `  T )  =  ( s  e.  P  |->  [_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  /  a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
291, 28eqtri 2472 1  |-  R  =  ( s  e.  P  |-> 
[_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  / 
a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1383    e. wcel 1804   _Vcvv 3095   [_csb 3420    u. cun 3459    i^i cin 3460   (/)c0 3770   {csn 4014   <.cotp 4022   U.cuni 4234    |-> cmpt 4495    X. cxp 4987   "cima 4992   ` cfv 5578   1stc1st 6783   2ndc2nd 6784  mVarscmvrs 28807  mPreStcmpst 28811  mStRedcmsr 28812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-ot 4023  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-msr 28832
This theorem is referenced by:  msrval  28876  msrf  28880
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