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Theorem msrfval 30247
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 5879 . . . . . 6  |-  ( t  =  T  ->  (mPreSt `  t )  =  (mPreSt `  T ) )
3 msrfval.p . . . . . 6  |-  P  =  (mPreSt `  T )
42, 3syl6eqr 2523 . . . . 5  |-  ( t  =  T  ->  (mPreSt `  t )  =  P )
5 fveq2 5879 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  (mVars `  t )  =  (mVars `  T ) )
6 msrfval.v . . . . . . . . . . . . 13  |-  V  =  (mVars `  T )
75, 6syl6eqr 2523 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (mVars `  t )  =  V )
87imaeq1d 5173 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
(mVars `  t ) " ( h  u. 
{ a } ) )  =  ( V
" ( h  u. 
{ a } ) ) )
98unieqd 4200 . . . . . . . . . 10  |-  ( t  =  T  ->  U. (
(mVars `  t ) " ( h  u. 
{ a } ) )  =  U. ( V " ( h  u. 
{ a } ) ) )
109csbeq1d 3356 . . . . . . . . 9  |-  ( t  =  T  ->  [_ U. ( (mVars `  t ) " ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
)  =  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) )
1110ineq2d 3625 . . . . . . . 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 4170 . . . . . . 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 3785 . . . . . 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 3785 . . . . 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 4474 . . . 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 30204 . . . 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 5889 . . . . . 6  |-  (mPreSt `  T )  e.  _V
183, 17eqeltri 2545 . . . . 5  |-  P  e. 
_V
1918mptex 6152 . . . 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 5963 . . 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 5715 . . . . 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 2480 . . . 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 5873 . . . 4  |-  ( -.  T  e.  _V  ->  (mStRed `  T )  =  (/) )
24 fvprc 5873 . . . . . 6  |-  ( -.  T  e.  _V  ->  (mPreSt `  T )  =  (/) )
253, 24syl5eq 2517 . . . . 5  |-  ( -.  T  e.  _V  ->  P  =  (/) )
2625mpteq1d 4477 . . . 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 2531 . . 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 169 . 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 2493 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 1452    e. wcel 1904   _Vcvv 3031   [_csb 3349    u. cun 3388    i^i cin 3389   (/)c0 3722   {csn 3959   <.cotp 3967   U.cuni 4190    |-> cmpt 4454    X. cxp 4837   "cima 4842   ` cfv 5589   1stc1st 6810   2ndc2nd 6811  mVarscmvrs 30179  mPreStcmpst 30183  mStRedcmsr 30184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-ot 3968  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-msr 30204
This theorem is referenced by:  msrval  30248  msrf  30252
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