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Theorem msrfval 30177
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 2482 . . . . 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 2482 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (mVars `  t )  =  V )
87imaeq1d 5184 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
(mVars `  t ) " ( h  u. 
{ a } ) )  =  ( V
" ( h  u. 
{ a } ) ) )
98unieqd 4227 . . . . . . . . . 10  |-  ( t  =  T  ->  U. (
(mVars `  t ) " ( h  u. 
{ a } ) )  =  U. ( V " ( h  u. 
{ a } ) ) )
109csbeq1d 3403 . . . . . . . . 9  |-  ( t  =  T  ->  [_ U. ( (mVars `  t ) " ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
)  =  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) )
1110ineq2d 3665 . . . . . . . 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 4197 . . . . . . 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 3810 . . . . . 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 3810 . . . . 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 4500 . . . 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 30134 . . . 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 2507 . . . . 5  |-  P  e. 
_V
1918mptex 6149 . . . 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 5962 . . 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 5721 . . . . 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 2436 . . . 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 2476 . . . . 5  |-  ( -.  T  e.  _V  ->  P  =  (/) )
2625mpteq1d 4503 . . . 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 2490 . . 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 168 . 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 2452 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 1438    e. wcel 1869   _Vcvv 3082   [_csb 3396    u. cun 3435    i^i cin 3436   (/)c0 3762   {csn 3997   <.cotp 4005   U.cuni 4217    |-> cmpt 4480    X. cxp 4849   "cima 4854   ` cfv 5599   1stc1st 6803   2ndc2nd 6804  mVarscmvrs 30109  mPreStcmpst 30113  mStRedcmsr 30114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-ot 4006  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-msr 30134
This theorem is referenced by:  msrval  30178  msrf  30182
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