Theorem msrid 31442

Description: The reduct of a statement is itself. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mstaval.r	|- R = (mStRed ` T )
mstaval.s	|- S = (mStat ` T )

Assertion

Ref	Expression
msrid	|- ( X e. S -> ( R ` X ) = X )

Proof of Theorem msrid

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- (mPreSt ` T ) = (mPreSt ` T )
2		mstaval.r	. . . . 5 |- R = (mStRed ` T )
3	1, 2	msrf 31439	. . . 4 |- R : (mPreSt ` T ) --> (mPreSt ` T )
4		ffn 6045	. . . 4 |- ( R : (mPreSt ` T ) --> (mPreSt ` T ) -> R Fn (mPreSt ` T ) )
5		fvelrnb 6243	. . . 4 |- ( R Fn (mPreSt ` T ) -> ( X e. ran R <-> E. s e. (mPreSt ` T ) ( R ` s ) = X ) )
6	3, 4, 5	mp2b 10	. . 3 |- ( X e. ran R <-> E. s e. (mPreSt ` T ) ( R ` s ) = X )
7	1	mpst123 31437	. . . . . . . . . . 11 |- ( s e. (mPreSt ` T ) -> s = <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. )
8	7	fveq2d 6195	. . . . . . . . . 10 |- ( s e. (mPreSt ` T ) -> ( R ` s ) = ( R ` <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) )
9		id 22	. . . . . . . . . . . 12 |- ( s e. (mPreSt ` T ) -> s e. (mPreSt ` T ) )
10	7, 9	eqeltrrd 2702	. . . . . . . . . . 11 |- ( s e. (mPreSt ` T ) -> <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. e. (mPreSt ` T ) )
11		eqid 2622	. . . . . . . . . . . 12 |- (mVars ` T ) = (mVars ` T )
12		eqid 2622	. . . . . . . . . . . 12 |- U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) = U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) )
13	11, 1, 2, 12	msrval 31435	. . . . . . . . . . 11 |- ( <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. e. (mPreSt ` T ) -> ( R ` <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) = <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. )
14	10, 13	syl 17	. . . . . . . . . 10 |- ( s e. (mPreSt ` T ) -> ( R ` <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) = <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. )
15	8, 14	eqtrd 2656	. . . . . . . . 9 |- ( s e. (mPreSt ` T ) -> ( R ` s ) = <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. )
16	3	ffvelrni 6358	. . . . . . . . 9 |- ( s e. (mPreSt ` T ) -> ( R ` s ) e. (mPreSt ` T ) )
17	15, 16	eqeltrrd 2702	. . . . . . . 8 |- ( s e. (mPreSt ` T ) -> <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. e. (mPreSt ` T ) )
18	11, 1, 2, 12	msrval 31435	. . . . . . . 8 |- ( <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. e. (mPreSt ` T ) -> ( R ` <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) = <. ( ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. )
19	17, 18	syl 17	. . . . . . 7 |- ( s e. (mPreSt ` T ) -> ( R ` <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) = <. ( ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. )
20		inass 3823	. . . . . . . . . 10 |- ( ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) = ( ( 1st ` ( 1st ` s ) ) i^i ( ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) )
21		inidm 3822	. . . . . . . . . . 11 |- ( ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) = ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) )
22	21	ineq2i 3811	. . . . . . . . . 10 |- ( ( 1st ` ( 1st ` s ) ) i^i ( ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) ) = ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) )
23	20, 22	eqtri 2644	. . . . . . . . 9 |- ( ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) = ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) )
24	23	a1i 11	. . . . . . . 8 |- ( s e. (mPreSt ` T ) -> ( ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) = ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) )
25	24	oteq1d 4414	. . . . . . 7 |- ( s e. (mPreSt ` T ) -> <. ( ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. = <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. )
26	19, 25	eqtrd 2656	. . . . . 6 |- ( s e. (mPreSt ` T ) -> ( R ` <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) = <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. )
27	15	fveq2d 6195	. . . . . 6 |- ( s e. (mPreSt ` T ) -> ( R ` ( R ` s ) ) = ( R ` <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) )
28	26, 27, 15	3eqtr4d 2666	. . . . 5 |- ( s e. (mPreSt ` T ) -> ( R ` ( R ` s ) ) = ( R ` s ) )
29		fveq2 6191	. . . . . 6 |- ( ( R ` s ) = X -> ( R ` ( R ` s ) ) = ( R ` X ) )
30		id 22	. . . . . 6 |- ( ( R ` s ) = X -> ( R ` s ) = X )
31	29, 30	eqeq12d 2637	. . . . 5 |- ( ( R ` s ) = X -> ( ( R ` ( R ` s ) ) = ( R ` s ) <-> ( R ` X ) = X ) )
32	28, 31	syl5ibcom 235	. . . 4 |- ( s e. (mPreSt ` T ) -> ( ( R ` s ) = X -> ( R ` X ) = X ) )
33	32	rexlimiv 3027	. . 3 |- ( E. s e. (mPreSt ` T ) ( R ` s ) = X -> ( R ` X ) = X )
34	6, 33	sylbi 207	. 2 |- ( X e. ran R -> ( R ` X ) = X )
35		mstaval.s	. . 3 |- S = (mStat ` T )
36	2, 35	mstaval 31441	. 2 |- S = ran R
37	34, 36	eleq2s 2719	1 |- ( X e. S -> ( R ` X ) = X )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   E.wrex 2913   u. cun 3572   i^i cin 3573   {csn 4177   <.cotp 4185   U.cuni 4436   X. cxp 5112   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888   1stc1st 7166   2ndc2nd 7167  mVarscmvrs 31366  mPreStcmpst 31370  mStRedcmsr 31371  mStatcmsta 31372

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-fal 1489  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  df-msta 31392

This theorem is referenced by:  elmsta  31445