Theorem elmsta 31445

Description: Property of being a statement. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mstapst.p	|- P = (mPreSt ` T )
mstapst.s	|- S = (mStat ` T )
elmsta.v	|- V = (mVars ` T )
elmsta.z	|- Z = U. ( V " ( H u. { A } ) )

Assertion

Ref	Expression
elmsta	|- ( <. D , H , A >. e. S <-> ( <. D , H , A >. e. P /\ D C_ ( Z X. Z ) ) )

Proof of Theorem elmsta

Step	Hyp	Ref	Expression
1		mstapst.p	. . . . 5 |- P = (mPreSt ` T )
2		mstapst.s	. . . . 5 |- S = (mStat ` T )
3	1, 2	mstapst 31444	. . . 4 |- S C_ P
4	3	sseli 3599	. . 3 |- ( <. D , H , A >. e. S -> <. D , H , A >. e. P )
5		elmsta.v	. . . . . . . . . 10 |- V = (mVars ` T )
6		eqid 2622	. . . . . . . . . 10 |- (mStRed ` T ) = (mStRed ` T )
7		elmsta.z	. . . . . . . . . 10 |- Z = U. ( V " ( H u. { A } ) )
8	5, 1, 6, 7	msrval 31435	. . . . . . . . 9 |- ( <. D , H , A >. e. P -> ( (mStRed ` T ) ` <. D , H , A >. ) = <. ( D i^i ( Z X. Z ) ) , H , A >. )
9	4, 8	syl 17	. . . . . . . 8 |- ( <. D , H , A >. e. S -> ( (mStRed ` T ) ` <. D , H , A >. ) = <. ( D i^i ( Z X. Z ) ) , H , A >. )
10	6, 2	msrid 31442	. . . . . . . 8 |- ( <. D , H , A >. e. S -> ( (mStRed ` T ) ` <. D , H , A >. ) = <. D , H , A >. )
11	9, 10	eqtr3d 2658	. . . . . . 7 |- ( <. D , H , A >. e. S -> <. ( D i^i ( Z X. Z ) ) , H , A >. = <. D , H , A >. )
12	11	fveq2d 6195	. . . . . 6 |- ( <. D , H , A >. e. S -> ( 1st ` <. ( D i^i ( Z X. Z ) ) , H , A >. ) = ( 1st ` <. D , H , A >. ) )
13	12	fveq2d 6195	. . . . 5 |- ( <. D , H , A >. e. S -> ( 1st ` ( 1st ` <. ( D i^i ( Z X. Z ) ) , H , A >. ) ) = ( 1st ` ( 1st ` <. D , H , A >. ) ) )
14		inss1 3833	. . . . . . 7 |- ( D i^i ( Z X. Z ) ) C_ D
15	1	mpstrcl 31438	. . . . . . . . 9 |- ( <. D , H , A >. e. P -> ( D e. _V /\ H e. _V /\ A e. _V ) )
16	4, 15	syl 17	. . . . . . . 8 |- ( <. D , H , A >. e. S -> ( D e. _V /\ H e. _V /\ A e. _V ) )
17	16	simp1d 1073	. . . . . . 7 |- ( <. D , H , A >. e. S -> D e. _V )
18		ssexg 4804	. . . . . . 7 |- ( ( ( D i^i ( Z X. Z ) ) C_ D /\ D e. _V ) -> ( D i^i ( Z X. Z ) ) e. _V )
19	14, 17, 18	sylancr 695	. . . . . 6 |- ( <. D , H , A >. e. S -> ( D i^i ( Z X. Z ) ) e. _V )
20	16	simp2d 1074	. . . . . 6 |- ( <. D , H , A >. e. S -> H e. _V )
21	16	simp3d 1075	. . . . . 6 |- ( <. D , H , A >. e. S -> A e. _V )
22		ot1stg 7182	. . . . . 6 |- ( ( ( D i^i ( Z X. Z ) ) e. _V /\ H e. _V /\ A e. _V ) -> ( 1st ` ( 1st ` <. ( D i^i ( Z X. Z ) ) , H , A >. ) ) = ( D i^i ( Z X. Z ) ) )
23	19, 20, 21, 22	syl3anc 1326	. . . . 5 |- ( <. D , H , A >. e. S -> ( 1st ` ( 1st ` <. ( D i^i ( Z X. Z ) ) , H , A >. ) ) = ( D i^i ( Z X. Z ) ) )
24		ot1stg 7182	. . . . . 6 |- ( ( D e. _V /\ H e. _V /\ A e. _V ) -> ( 1st ` ( 1st ` <. D , H , A >. ) ) = D )
25	16, 24	syl 17	. . . . 5 |- ( <. D , H , A >. e. S -> ( 1st ` ( 1st ` <. D , H , A >. ) ) = D )
26	13, 23, 25	3eqtr3d 2664	. . . 4 |- ( <. D , H , A >. e. S -> ( D i^i ( Z X. Z ) ) = D )
27		inss2 3834	. . . 4 |- ( D i^i ( Z X. Z ) ) C_ ( Z X. Z )
28	26, 27	syl6eqssr 3656	. . 3 |- ( <. D , H , A >. e. S -> D C_ ( Z X. Z ) )
29	4, 28	jca 554	. 2 |- ( <. D , H , A >. e. S -> ( <. D , H , A >. e. P /\ D C_ ( Z X. Z ) ) )
30	8	adantr 481	. . . . 5 |- ( ( <. D , H , A >. e. P /\ D C_ ( Z X. Z ) ) -> ( (mStRed ` T ) ` <. D , H , A >. ) = <. ( D i^i ( Z X. Z ) ) , H , A >. )
31		simpr 477	. . . . . . 7 |- ( ( <. D , H , A >. e. P /\ D C_ ( Z X. Z ) ) -> D C_ ( Z X. Z ) )
32		df-ss 3588	. . . . . . 7 |- ( D C_ ( Z X. Z ) <-> ( D i^i ( Z X. Z ) ) = D )
33	31, 32	sylib 208	. . . . . 6 |- ( ( <. D , H , A >. e. P /\ D C_ ( Z X. Z ) ) -> ( D i^i ( Z X. Z ) ) = D )
34	33	oteq1d 4414	. . . . 5 |- ( ( <. D , H , A >. e. P /\ D C_ ( Z X. Z ) ) -> <. ( D i^i ( Z X. Z ) ) , H , A >. = <. D , H , A >. )
35	30, 34	eqtrd 2656	. . . 4 |- ( ( <. D , H , A >. e. P /\ D C_ ( Z X. Z ) ) -> ( (mStRed ` T ) ` <. D , H , A >. ) = <. D , H , A >. )
36	1, 6	msrf 31439	. . . . . 6 |- (mStRed ` T ) : P --> P
37		ffn 6045	. . . . . 6 |- ( (mStRed ` T ) : P --> P -> (mStRed ` T ) Fn P )
38	36, 37	ax-mp 5	. . . . 5 |- (mStRed ` T ) Fn P
39		simpl 473	. . . . 5 |- ( ( <. D , H , A >. e. P /\ D C_ ( Z X. Z ) ) -> <. D , H , A >. e. P )
40		fnfvelrn 6356	. . . . 5 |- ( ( (mStRed ` T ) Fn P /\ <. D , H , A >. e. P ) -> ( (mStRed ` T ) ` <. D , H , A >. ) e. ran (mStRed ` T ) )
41	38, 39, 40	sylancr 695	. . . 4 |- ( ( <. D , H , A >. e. P /\ D C_ ( Z X. Z ) ) -> ( (mStRed ` T ) ` <. D , H , A >. ) e. ran (mStRed ` T ) )
42	35, 41	eqeltrrd 2702	. . 3 |- ( ( <. D , H , A >. e. P /\ D C_ ( Z X. Z ) ) -> <. D , H , A >. e. ran (mStRed ` T ) )
43	6, 2	mstaval 31441	. . 3 |- S = ran (mStRed ` T )
44	42, 43	syl6eleqr 2712	. 2 |- ( ( <. D , H , A >. e. P /\ D C_ ( Z X. Z ) ) -> <. D , H , A >. e. S )
45	29, 44	impbii 199	1 |- ( <. D , H , A >. e. S <-> ( <. D , H , A >. e. P /\ D C_ ( Z X. Z ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   {csn 4177   <.cotp 4185   U.cuni 4436   X. cxp 5112   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888   1stc1st 7166  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: (None)