Theorem msrfval 31434

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.

Step	Hyp	Ref	Expression
1		msrfval.r	. 2 |- R = (mStRed ` T )
2		fveq2 6191	. . . . . 6 |- ( t = T -> (mPreSt ` t ) = (mPreSt ` T ) )
3		msrfval.p	. . . . . 6 |- P = (mPreSt ` T )
4	2, 3	syl6eqr 2674	. . . . 5 |- ( t = T -> (mPreSt ` t ) = P )
5		fveq2 6191	. . . . . . . . . . . . 13 |- ( t = T -> (mVars ` t ) = (mVars ` T ) )
6		msrfval.v	. . . . . . . . . . . . 13 |- V = (mVars ` T )
7	5, 6	syl6eqr 2674	. . . . . . . . . . . 12 |- ( t = T -> (mVars ` t ) = V )
8	7	imaeq1d 5465	. . . . . . . . . . 11 |- ( t = T -> ( (mVars ` t ) " ( h u. { a } ) ) = ( V " ( h u. { a } ) ) )
9	8	unieqd 4446	. . . . . . . . . 10 |- ( t = T -> U. ( (mVars ` t ) " ( h u. { a } ) ) = U. ( V " ( h u. { a } ) ) )
10	9	csbeq1d 3540	. . . . . . . . 9 |- ( t = T -> [_ U. ( (mVars ` t ) " ( h u. { a } ) ) / z ]_ ( z X. z ) = [_ U. ( V " ( h u. { a } ) ) / z ]_ ( z X. z ) )
11	10	ineq2d 3814	. . . . . . . 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 ) ) )
12	11	oteq1d 4414	. . . . . . 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 >. )
13	12	csbeq2dv 3992	. . . . . 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 >. )
14	13	csbeq2dv 3992	. . . . 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 >. )
15	4, 14	mpteq12dv 4733	. . . 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 31391	. . . 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 6201	. . . . . 6 |- (mPreSt ` T ) e. _V
18	3, 17	eqeltri 2697	. . . . 5 |- P e. _V
19	18	mptex 6486	. . . 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
20	15, 16, 19	fvmpt 6282	. . 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 6021	. . . . 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 >. ) = (/)
22	21	eqcomi 2631	. . . 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 6185	. . . 4 |- ( -. T e. _V -> (mStRed ` T ) = (/) )
24		fvprc 6185	. . . . . 6 |- ( -. T e. _V -> (mPreSt ` T ) = (/) )
25	3, 24	syl5eq 2668	. . . . 5 |- ( -. T e. _V -> P = (/) )
26	25	mpteq1d 4738	. . . 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 >. ) )
27	22, 23, 26	3eqtr4a 2682	. . 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 >. ) )
28	20, 27	pm2.61i 176	. 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 >. )
29	1, 28	eqtri 2644	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 >. )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   <.cotp 4185   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   "cima 5117   ` cfv 5888   1stc1st 7166   2ndc2nd 7167  mVarscmvrs 31366  mPreStcmpst 31370  mStRedcmsr 31371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-msr 31391

This theorem is referenced by:  msrval  31435  msrf  31439