Theorem msrf 31439

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

Hypotheses

Ref	Expression
mpstssv.p	|- P = (mPreSt ` T )
msrf.r	|- R = (mStRed ` T )

Assertion

Ref	Expression
msrf	|- R : P --> P

Proof of Theorem msrf

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

Step	Hyp	Ref	Expression
1		otex 4933	. . . . 5 |- <. ( ( 1st ` ( 1st ` s ) ) i^i [_ U. ( (mVars ` T ) " ( h u. { a } ) ) / z ]_ ( z X. z ) ) , h , a >. e. _V
2	1	csbex 4793	. . . 4 |- [_ ( 2nd ` s ) / a ]_ <. ( ( 1st ` ( 1st ` s ) ) i^i [_ U. ( (mVars ` T ) " ( h u. { a } ) ) / z ]_ ( z X. z ) ) , h , a >. e. _V
3	2	csbex 4793	. . 3 |- [_ ( 2nd ` ( 1st ` s ) ) / h ]_ [_ ( 2nd ` s ) / a ]_ <. ( ( 1st ` ( 1st ` s ) ) i^i [_ U. ( (mVars ` T ) " ( h u. { a } ) ) / z ]_ ( z X. z ) ) , h , a >. e. _V
4		eqid 2622	. . . 4 |- (mVars ` T ) = (mVars ` T )
5		mpstssv.p	. . . 4 |- P = (mPreSt ` T )
6		msrf.r	. . . 4 |- R = (mStRed ` T )
7	4, 5, 6	msrfval 31434	. . 3 |- R = ( s e. P |-> [_ ( 2nd ` ( 1st ` s ) ) / h ]_ [_ ( 2nd ` s ) / a ]_ <. ( ( 1st ` ( 1st ` s ) ) i^i [_ U. ( (mVars ` T ) " ( h u. { a } ) ) / z ]_ ( z X. z ) ) , h , a >. )
8	3, 7	fnmpti 6022	. 2 |- R Fn P
9	5	mpst123 31437	. . . . . 6 |- ( s e. P -> s = <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. )
10	9	fveq2d 6195	. . . . 5 |- ( s e. P -> ( R ` s ) = ( R ` <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) )
11		id 22	. . . . . . 7 |- ( s e. P -> s e. P )
12	9, 11	eqeltrrd 2702	. . . . . 6 |- ( s e. P -> <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. e. P )
13		eqid 2622	. . . . . . 7 |- U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) = U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) )
14	4, 5, 6, 13	msrval 31435	. . . . . 6 |- ( <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. e. P -> ( 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	12, 14	syl 17	. . . . 5 |- ( s e. P -> ( 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 ) >. )
16	10, 15	eqtrd 2656	. . . 4 |- ( s e. P -> ( 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 ) >. )
17		inss1 3833	. . . . . . 7 |- ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) C_ ( 1st ` ( 1st ` s ) )
18		eqid 2622	. . . . . . . . . . 11 |- (mDV ` T ) = (mDV ` T )
19		eqid 2622	. . . . . . . . . . 11 |- (mEx ` T ) = (mEx ` T )
20	18, 19, 5	elmpst 31433	. . . . . . . . . 10 |- ( <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. e. P <-> ( ( ( 1st ` ( 1st ` s ) ) C_ (mDV ` T ) /\ `' ( 1st ` ( 1st ` s ) ) = ( 1st ` ( 1st ` s ) ) ) /\ ( ( 2nd ` ( 1st ` s ) ) C_ (mEx ` T ) /\ ( 2nd ` ( 1st ` s ) ) e. Fin ) /\ ( 2nd ` s ) e. (mEx ` T ) ) )
21	12, 20	sylib 208	. . . . . . . . 9 |- ( s e. P -> ( ( ( 1st ` ( 1st ` s ) ) C_ (mDV ` T ) /\ `' ( 1st ` ( 1st ` s ) ) = ( 1st ` ( 1st ` s ) ) ) /\ ( ( 2nd ` ( 1st ` s ) ) C_ (mEx ` T ) /\ ( 2nd ` ( 1st ` s ) ) e. Fin ) /\ ( 2nd ` s ) e. (mEx ` T ) ) )
22	21	simp1d 1073	. . . . . . . 8 |- ( s e. P -> ( ( 1st ` ( 1st ` s ) ) C_ (mDV ` T ) /\ `' ( 1st ` ( 1st ` s ) ) = ( 1st ` ( 1st ` s ) ) ) )
23	22	simpld 475	. . . . . . 7 |- ( s e. P -> ( 1st ` ( 1st ` s ) ) C_ (mDV ` T ) )
24	17, 23	syl5ss 3614	. . . . . 6 |- ( s e. P -> ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) C_ (mDV ` T ) )
25		cnvin 5540	. . . . . . 7 |- `' ( ( 1st ` ( 1st ` 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 ) } ) ) ) )
26	22	simprd 479	. . . . . . . 8 |- ( s e. P -> `' ( 1st ` ( 1st ` s ) ) = ( 1st ` ( 1st ` s ) ) )
27		cnvxp 5551	. . . . . . . . 9 |- `' ( 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 ) } ) ) )
28	27	a1i 11	. . . . . . . 8 |- ( s e. P -> `' ( 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 ) } ) ) ) )
29	26, 28	ineq12d 3815	. . . . . . 7 |- ( s e. P -> ( `' ( 1st ` ( 1st ` 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 ) } ) ) ) ) )
30	25, 29	syl5eq 2668	. . . . . 6 |- ( s e. P -> `' ( ( 1st ` ( 1st ` 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 ) } ) ) ) ) )
31	24, 30	jca 554	. . . . 5 |- ( s e. P -> ( ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) C_ (mDV ` 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 ) } ) ) ) ) = ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) ) )
32	21	simp2d 1074	. . . . 5 |- ( s e. P -> ( ( 2nd ` ( 1st ` s ) ) C_ (mEx ` T ) /\ ( 2nd ` ( 1st ` s ) ) e. Fin ) )
33	21	simp3d 1075	. . . . 5 |- ( s e. P -> ( 2nd ` s ) e. (mEx ` T ) )
34	18, 19, 5	elmpst 31433	. . . . 5 |- ( <. ( ( 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. P <-> ( ( ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( (mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) C_ (mDV ` 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 ) } ) ) ) ) = ( ( 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 ) ) C_ (mEx ` T ) /\ ( 2nd ` ( 1st ` s ) ) e. Fin ) /\ ( 2nd ` s ) e. (mEx ` T ) ) )
35	31, 32, 33, 34	syl3anbrc 1246	. . . 4 |- ( s e. P -> <. ( ( 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. P )
36	16, 35	eqeltrd 2701	. . 3 |- ( s e. P -> ( R ` s ) e. P )
37	36	rgen 2922	. 2 |- A. s e. P ( R ` s ) e. P
38		ffnfv 6388	. 2 |- ( R : P --> P <-> ( R Fn P /\ A. s e. P ( R ` s ) e. P ) )
39	8, 37, 38	mpbir2an 955	1 |- R : P --> P

Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   [_csb 3533   u. cun 3572   i^i cin 3573   C_ wss 3574   {csn 4177   <.cotp 4185   U.cuni 4436   X. cxp 5112   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` 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-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

This theorem is referenced by:  msrrcl  31440  msrid  31442  msrfo  31443  mstapst  31444  elmsta  31445  elmthm  31473  mthmsta  31475  mthmblem  31477