Theorem msrval 31435

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 )
msrval.z	|- Z = U. ( V " ( H u. { A } ) )

Assertion

Ref	Expression
msrval	|- ( <. D , H , A >. e. P -> ( R ` <. D , H , A >. ) = <. ( D i^i ( Z X. Z ) ) , H , A >. )

Proof of Theorem msrval

Dummy variables h a s z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		msrfval.v	. . . 4 |- V = (mVars ` T )
2		msrfval.p	. . . 4 |- P = (mPreSt ` T )
3		msrfval.r	. . . 4 |- R = (mStRed ` T )
4	1, 2, 3	msrfval 31434	. . 3 |- 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 >. )
5	4	a1i 11	. 2 |- ( <. D , H , A >. e. P -> 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 >. ) )
6		fvexd 6203	. . 3 |- ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) -> ( 2nd ` ( 1st ` s ) ) e. _V )
7		fvexd 6203	. . . 4 |- ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) -> ( 2nd ` s ) e. _V )
8		simpllr 799	. . . . . . . . 9 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> s = <. D , H , A >. )
9	8	fveq2d 6195	. . . . . . . 8 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> ( 1st ` s ) = ( 1st ` <. D , H , A >. ) )
10	9	fveq2d 6195	. . . . . . 7 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> ( 1st ` ( 1st ` s ) ) = ( 1st ` ( 1st ` <. D , H , A >. ) ) )
11		eqid 2622	. . . . . . . . . . . . 13 |- (mDV ` T ) = (mDV ` T )
12		eqid 2622	. . . . . . . . . . . . 13 |- (mEx ` T ) = (mEx ` T )
13	11, 12, 2	elmpst 31433	. . . . . . . . . . . 12 |- ( <. D , H , A >. e. P <-> ( ( D C_ (mDV ` T ) /\ `' D = D ) /\ ( H C_ (mEx ` T ) /\ H e. Fin ) /\ A e. (mEx ` T ) ) )
14	13	simp1bi 1076	. . . . . . . . . . 11 |- ( <. D , H , A >. e. P -> ( D C_ (mDV ` T ) /\ `' D = D ) )
15	14	simpld 475	. . . . . . . . . 10 |- ( <. D , H , A >. e. P -> D C_ (mDV ` T ) )
16	15	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> D C_ (mDV ` T ) )
17		fvex 6201	. . . . . . . . . 10 |- (mDV ` T ) e. _V
18	17	ssex 4802	. . . . . . . . 9 |- ( D C_ (mDV ` T ) -> D e. _V )
19	16, 18	syl 17	. . . . . . . 8 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> D e. _V )
20	13	simp2bi 1077	. . . . . . . . . 10 |- ( <. D , H , A >. e. P -> ( H C_ (mEx ` T ) /\ H e. Fin ) )
21	20	simprd 479	. . . . . . . . 9 |- ( <. D , H , A >. e. P -> H e. Fin )
22	21	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> H e. Fin )
23	13	simp3bi 1078	. . . . . . . . 9 |- ( <. D , H , A >. e. P -> A e. (mEx ` T ) )
24	23	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> A e. (mEx ` T ) )
25		ot1stg 7182	. . . . . . . 8 |- ( ( D e. _V /\ H e. Fin /\ A e. (mEx ` T ) ) -> ( 1st ` ( 1st ` <. D , H , A >. ) ) = D )
26	19, 22, 24, 25	syl3anc 1326	. . . . . . 7 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> ( 1st ` ( 1st ` <. D , H , A >. ) ) = D )
27	10, 26	eqtrd 2656	. . . . . 6 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> ( 1st ` ( 1st ` s ) ) = D )
28		fvex 6201	. . . . . . . . . . 11 |- (mVars ` T ) e. _V
29	1, 28	eqeltri 2697	. . . . . . . . . 10 |- V e. _V
30		imaexg 7103	. . . . . . . . . 10 |- ( V e. _V -> ( V " ( h u. { a } ) ) e. _V )
31	29, 30	ax-mp 5	. . . . . . . . 9 |- ( V " ( h u. { a } ) ) e. _V
32	31	uniex 6953	. . . . . . . 8 |- U. ( V " ( h u. { a } ) ) e. _V
33	32	a1i 11	. . . . . . 7 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> U. ( V " ( h u. { a } ) ) e. _V )
34		id 22	. . . . . . . . 9 |- ( z = U. ( V " ( h u. { a } ) ) -> z = U. ( V " ( h u. { a } ) ) )
35		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> h = ( 2nd ` ( 1st ` s ) ) )
36	9	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> ( 2nd ` ( 1st ` s ) ) = ( 2nd ` ( 1st ` <. D , H , A >. ) ) )
37		ot2ndg 7183	. . . . . . . . . . . . . . 15 |- ( ( D e. _V /\ H e. Fin /\ A e. (mEx ` T ) ) -> ( 2nd ` ( 1st ` <. D , H , A >. ) ) = H )
38	19, 22, 24, 37	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> ( 2nd ` ( 1st ` <. D , H , A >. ) ) = H )
39	35, 36, 38	3eqtrd 2660	. . . . . . . . . . . . 13 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> h = H )
40		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> a = ( 2nd ` s ) )
41	8	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> ( 2nd ` s ) = ( 2nd ` <. D , H , A >. ) )
42		ot3rdg 7184	. . . . . . . . . . . . . . . 16 |- ( A e. (mEx ` T ) -> ( 2nd ` <. D , H , A >. ) = A )
43	24, 42	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> ( 2nd ` <. D , H , A >. ) = A )
44	40, 41, 43	3eqtrd 2660	. . . . . . . . . . . . . 14 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> a = A )
45	44	sneqd 4189	. . . . . . . . . . . . 13 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> { a } = { A } )
46	39, 45	uneq12d 3768	. . . . . . . . . . . 12 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> ( h u. { a } ) = ( H u. { A } ) )
47	46	imaeq2d 5466	. . . . . . . . . . 11 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> ( V " ( h u. { a } ) ) = ( V " ( H u. { A } ) ) )
48	47	unieqd 4446	. . . . . . . . . 10 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> U. ( V " ( h u. { a } ) ) = U. ( V " ( H u. { A } ) ) )
49		msrval.z	. . . . . . . . . 10 |- Z = U. ( V " ( H u. { A } ) )
50	48, 49	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> U. ( V " ( h u. { a } ) ) = Z )
51	34, 50	sylan9eqr 2678	. . . . . . . 8 |- ( ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) /\ z = U. ( V " ( h u. { a } ) ) ) -> z = Z )
52	51	sqxpeqd 5141	. . . . . . 7 |- ( ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) /\ z = U. ( V " ( h u. { a } ) ) ) -> ( z X. z ) = ( Z X. Z ) )
53	33, 52	csbied 3560	. . . . . 6 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> [_ U. ( V " ( h u. { a } ) ) / z ]_ ( z X. z ) = ( Z X. Z ) )
54	27, 53	ineq12d 3815	. . . . 5 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> ( ( 1st ` ( 1st ` s ) ) i^i [_ U. ( V " ( h u. { a } ) ) / z ]_ ( z X. z ) ) = ( D i^i ( Z X. Z ) ) )
55	54, 39, 44	oteq123d 4417	. . . 4 |- ( ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) /\ a = ( 2nd ` s ) ) -> <. ( ( 1st ` ( 1st ` s ) ) i^i [_ U. ( V " ( h u. { a } ) ) / z ]_ ( z X. z ) ) , h , a >. = <. ( D i^i ( Z X. Z ) ) , H , A >. )
56	7, 55	csbied 3560	. . 3 |- ( ( ( <. D , H , A >. e. P /\ s = <. D , H , A >. ) /\ h = ( 2nd ` ( 1st ` s ) ) ) -> [_ ( 2nd ` s ) / a ]_ <. ( ( 1st ` ( 1st ` s ) ) i^i [_ U. ( V " ( h u. { a } ) ) / z ]_ ( z X. z ) ) , h , a >. = <. ( D i^i ( Z X. Z ) ) , H , A >. )
57	6, 56	csbied 3560	. 2 |- ( ( <. D , H , A >. e. P /\ s = <. D , 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 >. = <. ( D i^i ( Z X. Z ) ) , H , A >. )
58		id 22	. 2 |- ( <. D , H , A >. e. P -> <. D , H , A >. e. P )
59		otex 4933	. . 3 |- <. ( D i^i ( Z X. Z ) ) , H , A >. e. _V
60	59	a1i 11	. 2 |- ( <. D , H , A >. e. P -> <. ( D i^i ( Z X. Z ) ) , H , A >. e. _V )
61	5, 57, 58, 60	fvmptd 6288	1 |- ( <. D , H , A >. e. P -> ( R ` <. D , H , A >. ) = <. ( D i^i ( Z X. Z ) ) , H , A >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   u. cun 3572   i^i cin 3573   C_ wss 3574   {csn 4177   <.cotp 4185   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   "cima 5117   ` cfv 5888   1stc1st 7166   2ndc2nd 7167   Fincfn 7955  mExcmex 31364  mDVcmdv 31365  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  ax-un 6949

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-pw 4160  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-1st 7168  df-2nd 7169  df-mpst 31390  df-msr 31391

This theorem is referenced by:  msrf  31439  msrid  31442  elmsta  31445  mthmpps  31479