Theorem mthmpps 31479

Description: Given a theorem, there is an explicitly definable witnessing provable pre-statement for the provability of the theorem. (However, this pre-statement requires infinitely many dv conditions, which is sometimes inconvenient.) (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mthmpps.r	|- R = (mStRed ` T )
mthmpps.j	|- J = (mPPSt ` T )
mthmpps.u	|- U = (mThm ` T )
mthmpps.d	|- D = (mDV ` T )
mthmpps.v	|- V = (mVars ` T )
mthmpps.z	|- Z = U. ( V " ( H u. { A } ) )
mthmpps.m	|- M = ( C u. ( D \ ( Z X. Z ) ) )

Assertion

Ref	Expression
mthmpps	|- ( T e. mFS -> ( <. C , H , A >. e. U <-> ( <. M , H , A >. e. J /\ ( R ` <. M , H , A >. ) = ( R ` <. C , H , A >. ) ) ) )

Proof of Theorem mthmpps

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mthmpps.m	. . . . . . . 8 |- M = ( C u. ( D \ ( Z X. Z ) ) )
2		mthmpps.u	. . . . . . . . . . . . . 14 |- U = (mThm ` T )
3		eqid 2622	. . . . . . . . . . . . . 14 |- (mPreSt ` T ) = (mPreSt ` T )
4	2, 3	mthmsta 31475	. . . . . . . . . . . . 13 |- U C_ (mPreSt ` T )
5		simpr 477	. . . . . . . . . . . . 13 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> <. C , H , A >. e. U )
6	4, 5	sseldi 3601	. . . . . . . . . . . 12 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> <. C , H , A >. e. (mPreSt ` T ) )
7		mthmpps.d	. . . . . . . . . . . . 13 |- D = (mDV ` T )
8		eqid 2622	. . . . . . . . . . . . 13 |- (mEx ` T ) = (mEx ` T )
9	7, 8, 3	elmpst 31433	. . . . . . . . . . . 12 |- ( <. C , H , A >. e. (mPreSt ` T ) <-> ( ( C C_ D /\ `' C = C ) /\ ( H C_ (mEx ` T ) /\ H e. Fin ) /\ A e. (mEx ` T ) ) )
10	6, 9	sylib 208	. . . . . . . . . . 11 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> ( ( C C_ D /\ `' C = C ) /\ ( H C_ (mEx ` T ) /\ H e. Fin ) /\ A e. (mEx ` T ) ) )
11	10	simp1d 1073	. . . . . . . . . 10 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> ( C C_ D /\ `' C = C ) )
12	11	simpld 475	. . . . . . . . 9 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> C C_ D )
13		difssd 3738	. . . . . . . . 9 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> ( D \ ( Z X. Z ) ) C_ D )
14	12, 13	unssd 3789	. . . . . . . 8 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> ( C u. ( D \ ( Z X. Z ) ) ) C_ D )
15	1, 14	syl5eqss 3649	. . . . . . 7 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> M C_ D )
16	11	simprd 479	. . . . . . . . 9 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> `' C = C )
17		cnvdif 5539	. . . . . . . . . . 11 |- `' ( D \ ( Z X. Z ) ) = ( `' D \ `' ( Z X. Z ) )
18		cnvdif 5539	. . . . . . . . . . . . . 14 |- `' ( ( (mVR ` T ) X. (mVR ` T ) ) \ _I ) = ( `' ( (mVR ` T ) X. (mVR ` T ) ) \ `' _I )
19		cnvxp 5551	. . . . . . . . . . . . . . 15 |- `' ( (mVR ` T ) X. (mVR ` T ) ) = ( (mVR ` T ) X. (mVR ` T ) )
20		cnvi 5537	. . . . . . . . . . . . . . 15 |- `' _I = _I
21	19, 20	difeq12i 3726	. . . . . . . . . . . . . 14 |- ( `' ( (mVR ` T ) X. (mVR ` T ) ) \ `' _I ) = ( ( (mVR ` T ) X. (mVR ` T ) ) \ _I )
22	18, 21	eqtri 2644	. . . . . . . . . . . . 13 |- `' ( ( (mVR ` T ) X. (mVR ` T ) ) \ _I ) = ( ( (mVR ` T ) X. (mVR ` T ) ) \ _I )
23		eqid 2622	. . . . . . . . . . . . . . 15 |- (mVR ` T ) = (mVR ` T )
24	23, 7	mdvval 31401	. . . . . . . . . . . . . 14 |- D = ( ( (mVR ` T ) X. (mVR ` T ) ) \ _I )
25	24	cnveqi 5297	. . . . . . . . . . . . 13 |- `' D = `' ( ( (mVR ` T ) X. (mVR ` T ) ) \ _I )
26	22, 25, 24	3eqtr4i 2654	. . . . . . . . . . . 12 |- `' D = D
27		cnvxp 5551	. . . . . . . . . . . 12 |- `' ( Z X. Z ) = ( Z X. Z )
28	26, 27	difeq12i 3726	. . . . . . . . . . 11 |- ( `' D \ `' ( Z X. Z ) ) = ( D \ ( Z X. Z ) )
29	17, 28	eqtri 2644	. . . . . . . . . 10 |- `' ( D \ ( Z X. Z ) ) = ( D \ ( Z X. Z ) )
30	29	a1i 11	. . . . . . . . 9 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> `' ( D \ ( Z X. Z ) ) = ( D \ ( Z X. Z ) ) )
31	16, 30	uneq12d 3768	. . . . . . . 8 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> ( `' C u. `' ( D \ ( Z X. Z ) ) ) = ( C u. ( D \ ( Z X. Z ) ) ) )
32	1	cnveqi 5297	. . . . . . . . 9 |- `' M = `' ( C u. ( D \ ( Z X. Z ) ) )
33		cnvun 5538	. . . . . . . . 9 |- `' ( C u. ( D \ ( Z X. Z ) ) ) = ( `' C u. `' ( D \ ( Z X. Z ) ) )
34	32, 33	eqtri 2644	. . . . . . . 8 |- `' M = ( `' C u. `' ( D \ ( Z X. Z ) ) )
35	31, 34, 1	3eqtr4g 2681	. . . . . . 7 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> `' M = M )
36	15, 35	jca 554	. . . . . 6 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> ( M C_ D /\ `' M = M ) )
37	10	simp2d 1074	. . . . . 6 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> ( H C_ (mEx ` T ) /\ H e. Fin ) )
38	10	simp3d 1075	. . . . . 6 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> A e. (mEx ` T ) )
39	7, 8, 3	elmpst 31433	. . . . . 6 |- ( <. M , H , A >. e. (mPreSt ` T ) <-> ( ( M C_ D /\ `' M = M ) /\ ( H C_ (mEx ` T ) /\ H e. Fin ) /\ A e. (mEx ` T ) ) )
40	36, 37, 38, 39	syl3anbrc 1246	. . . . 5 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> <. M , H , A >. e. (mPreSt ` T ) )
41		mthmpps.r	. . . . . . . 8 |- R = (mStRed ` T )
42		mthmpps.j	. . . . . . . 8 |- J = (mPPSt ` T )
43	41, 42, 2	elmthm 31473	. . . . . . 7 |- ( <. C , H , A >. e. U <-> E. x e. J ( R ` x ) = ( R ` <. C , H , A >. ) )
44	5, 43	sylib 208	. . . . . 6 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> E. x e. J ( R ` x ) = ( R ` <. C , H , A >. ) )
45		eqid 2622	. . . . . . . 8 |- (mCls ` T ) = (mCls ` T )
46		simpll 790	. . . . . . . 8 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> T e. mFS )
47	15	adantr 481	. . . . . . . 8 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> M C_ D )
48	37	simpld 475	. . . . . . . . 9 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> H C_ (mEx ` T ) )
49	48	adantr 481	. . . . . . . 8 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> H C_ (mEx ` T ) )
50	3, 42	mppspst 31471	. . . . . . . . . . . . . . . . . . 19 |- J C_ (mPreSt ` T )
51		simprl 794	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> x e. J )
52	50, 51	sseldi 3601	. . . . . . . . . . . . . . . . . 18 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> x e. (mPreSt ` T ) )
53	3	mpst123 31437	. . . . . . . . . . . . . . . . . 18 |- ( x e. (mPreSt ` T ) -> x = <. ( 1st ` ( 1st ` x ) ) , ( 2nd ` ( 1st ` x ) ) , ( 2nd ` x ) >. )
54	52, 53	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> x = <. ( 1st ` ( 1st ` x ) ) , ( 2nd ` ( 1st ` x ) ) , ( 2nd ` x ) >. )
55	54	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( R ` x ) = ( R ` <. ( 1st ` ( 1st ` x ) ) , ( 2nd ` ( 1st ` x ) ) , ( 2nd ` x ) >. ) )
56		simprr 796	. . . . . . . . . . . . . . . 16 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( R ` x ) = ( R ` <. C , H , A >. ) )
57	55, 56	eqtr3d 2658	. . . . . . . . . . . . . . 15 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( R ` <. ( 1st ` ( 1st ` x ) ) , ( 2nd ` ( 1st ` x ) ) , ( 2nd ` x ) >. ) = ( R ` <. C , H , A >. ) )
58	54, 52	eqeltrrd 2702	. . . . . . . . . . . . . . . 16 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> <. ( 1st ` ( 1st ` x ) ) , ( 2nd ` ( 1st ` x ) ) , ( 2nd ` x ) >. e. (mPreSt ` T ) )
59		mthmpps.v	. . . . . . . . . . . . . . . . 17 |- V = (mVars ` T )
60		eqid 2622	. . . . . . . . . . . . . . . . 17 |- U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) = U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) )
61	59, 3, 41, 60	msrval 31435	. . . . . . . . . . . . . . . 16 |- ( <. ( 1st ` ( 1st ` x ) ) , ( 2nd ` ( 1st ` x ) ) , ( 2nd ` x ) >. e. (mPreSt ` T ) -> ( R ` <. ( 1st ` ( 1st ` x ) ) , ( 2nd ` ( 1st ` x ) ) , ( 2nd ` x ) >. ) = <. ( ( 1st ` ( 1st ` x ) ) i^i ( U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) X. U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) ) ) , ( 2nd ` ( 1st ` x ) ) , ( 2nd ` x ) >. )
62	58, 61	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( R ` <. ( 1st ` ( 1st ` x ) ) , ( 2nd ` ( 1st ` x ) ) , ( 2nd ` x ) >. ) = <. ( ( 1st ` ( 1st ` x ) ) i^i ( U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) X. U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) ) ) , ( 2nd ` ( 1st ` x ) ) , ( 2nd ` x ) >. )
63		mthmpps.z	. . . . . . . . . . . . . . . . . 18 |- Z = U. ( V " ( H u. { A } ) )
64	59, 3, 41, 63	msrval 31435	. . . . . . . . . . . . . . . . 17 |- ( <. C , H , A >. e. (mPreSt ` T ) -> ( R ` <. C , H , A >. ) = <. ( C i^i ( Z X. Z ) ) , H , A >. )
65	6, 64	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> ( R ` <. C , H , A >. ) = <. ( C i^i ( Z X. Z ) ) , H , A >. )
66	65	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( R ` <. C , H , A >. ) = <. ( C i^i ( Z X. Z ) ) , H , A >. )
67	57, 62, 66	3eqtr3d 2664	. . . . . . . . . . . . . 14 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> <. ( ( 1st ` ( 1st ` x ) ) i^i ( U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) X. U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) ) ) , ( 2nd ` ( 1st ` x ) ) , ( 2nd ` x ) >. = <. ( C i^i ( Z X. Z ) ) , H , A >. )
68		fvex 6201	. . . . . . . . . . . . . . . 16 |- ( 1st ` ( 1st ` x ) ) e. _V
69	68	inex1 4799	. . . . . . . . . . . . . . 15 |- ( ( 1st ` ( 1st ` x ) ) i^i ( U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) X. U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) ) ) e. _V
70		fvex 6201	. . . . . . . . . . . . . . 15 |- ( 2nd ` ( 1st ` x ) ) e. _V
71		fvex 6201	. . . . . . . . . . . . . . 15 |- ( 2nd ` x ) e. _V
72	69, 70, 71	otth 4953	. . . . . . . . . . . . . 14 |- ( <. ( ( 1st ` ( 1st ` x ) ) i^i ( U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) X. U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) ) ) , ( 2nd ` ( 1st ` x ) ) , ( 2nd ` x ) >. = <. ( C i^i ( Z X. Z ) ) , H , A >. <-> ( ( ( 1st ` ( 1st ` x ) ) i^i ( U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) X. U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) ) ) = ( C i^i ( Z X. Z ) ) /\ ( 2nd ` ( 1st ` x ) ) = H /\ ( 2nd ` x ) = A ) )
73	67, 72	sylib 208	. . . . . . . . . . . . 13 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( ( ( 1st ` ( 1st ` x ) ) i^i ( U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) X. U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) ) ) = ( C i^i ( Z X. Z ) ) /\ ( 2nd ` ( 1st ` x ) ) = H /\ ( 2nd ` x ) = A ) )
74	73	simp1d 1073	. . . . . . . . . . . 12 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( ( 1st ` ( 1st ` x ) ) i^i ( U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) X. U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) ) ) = ( C i^i ( Z X. Z ) ) )
75	73	simp2d 1074	. . . . . . . . . . . . . . . . . 18 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( 2nd ` ( 1st ` x ) ) = H )
76	73	simp3d 1075	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( 2nd ` x ) = A )
77	76	sneqd 4189	. . . . . . . . . . . . . . . . . 18 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> { ( 2nd ` x ) } = { A } )
78	75, 77	uneq12d 3768	. . . . . . . . . . . . . . . . 17 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) = ( H u. { A } ) )
79	78	imaeq2d 5466	. . . . . . . . . . . . . . . 16 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) = ( V " ( H u. { A } ) ) )
80	79	unieqd 4446	. . . . . . . . . . . . . . 15 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) = U. ( V " ( H u. { A } ) ) )
81	80, 63	syl6eqr 2674	. . . . . . . . . . . . . 14 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) = Z )
82	81	sqxpeqd 5141	. . . . . . . . . . . . 13 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) X. U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) ) = ( Z X. Z ) )
83	82	ineq2d 3814	. . . . . . . . . . . 12 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( ( 1st ` ( 1st ` x ) ) i^i ( U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) X. U. ( V " ( ( 2nd ` ( 1st ` x ) ) u. { ( 2nd ` x ) } ) ) ) ) = ( ( 1st ` ( 1st ` x ) ) i^i ( Z X. Z ) ) )
84	74, 83	eqtr3d 2658	. . . . . . . . . . 11 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( C i^i ( Z X. Z ) ) = ( ( 1st ` ( 1st ` x ) ) i^i ( Z X. Z ) ) )
85		inss1 3833	. . . . . . . . . . 11 |- ( C i^i ( Z X. Z ) ) C_ C
86	84, 85	syl6eqssr 3656	. . . . . . . . . 10 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( ( 1st ` ( 1st ` x ) ) i^i ( Z X. Z ) ) C_ C )
87		eqidd 2623	. . . . . . . . . . . . . . 15 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( 1st ` ( 1st ` x ) ) = ( 1st ` ( 1st ` x ) ) )
88	87, 75, 76	oteq123d 4417	. . . . . . . . . . . . . 14 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> <. ( 1st ` ( 1st ` x ) ) , ( 2nd ` ( 1st ` x ) ) , ( 2nd ` x ) >. = <. ( 1st ` ( 1st ` x ) ) , H , A >. )
89	54, 88	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> x = <. ( 1st ` ( 1st ` x ) ) , H , A >. )
90	89, 52	eqeltrrd 2702	. . . . . . . . . . . 12 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> <. ( 1st ` ( 1st ` x ) ) , H , A >. e. (mPreSt ` T ) )
91	7, 8, 3	elmpst 31433	. . . . . . . . . . . . . 14 |- ( <. ( 1st ` ( 1st ` x ) ) , H , A >. e. (mPreSt ` T ) <-> ( ( ( 1st ` ( 1st ` x ) ) C_ D /\ `' ( 1st ` ( 1st ` x ) ) = ( 1st ` ( 1st ` x ) ) ) /\ ( H C_ (mEx ` T ) /\ H e. Fin ) /\ A e. (mEx ` T ) ) )
92	91	simp1bi 1076	. . . . . . . . . . . . 13 |- ( <. ( 1st ` ( 1st ` x ) ) , H , A >. e. (mPreSt ` T ) -> ( ( 1st ` ( 1st ` x ) ) C_ D /\ `' ( 1st ` ( 1st ` x ) ) = ( 1st ` ( 1st ` x ) ) ) )
93	92	simpld 475	. . . . . . . . . . . 12 |- ( <. ( 1st ` ( 1st ` x ) ) , H , A >. e. (mPreSt ` T ) -> ( 1st ` ( 1st ` x ) ) C_ D )
94	90, 93	syl 17	. . . . . . . . . . 11 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( 1st ` ( 1st ` x ) ) C_ D )
95	94	ssdifd 3746	. . . . . . . . . 10 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( ( 1st ` ( 1st ` x ) ) \ ( Z X. Z ) ) C_ ( D \ ( Z X. Z ) ) )
96		unss12 3785	. . . . . . . . . 10 |- ( ( ( ( 1st ` ( 1st ` x ) ) i^i ( Z X. Z ) ) C_ C /\ ( ( 1st ` ( 1st ` x ) ) \ ( Z X. Z ) ) C_ ( D \ ( Z X. Z ) ) ) -> ( ( ( 1st ` ( 1st ` x ) ) i^i ( Z X. Z ) ) u. ( ( 1st ` ( 1st ` x ) ) \ ( Z X. Z ) ) ) C_ ( C u. ( D \ ( Z X. Z ) ) ) )
97	86, 95, 96	syl2anc 693	. . . . . . . . 9 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( ( ( 1st ` ( 1st ` x ) ) i^i ( Z X. Z ) ) u. ( ( 1st ` ( 1st ` x ) ) \ ( Z X. Z ) ) ) C_ ( C u. ( D \ ( Z X. Z ) ) ) )
98		inundif 4046	. . . . . . . . . 10 |- ( ( ( 1st ` ( 1st ` x ) ) i^i ( Z X. Z ) ) u. ( ( 1st ` ( 1st ` x ) ) \ ( Z X. Z ) ) ) = ( 1st ` ( 1st ` x ) )
99	98	eqcomi 2631	. . . . . . . . 9 |- ( 1st ` ( 1st ` x ) ) = ( ( ( 1st ` ( 1st ` x ) ) i^i ( Z X. Z ) ) u. ( ( 1st ` ( 1st ` x ) ) \ ( Z X. Z ) ) )
100	97, 99, 1	3sstr4g 3646	. . . . . . . 8 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( 1st ` ( 1st ` x ) ) C_ M )
101		ssid 3624	. . . . . . . . 9 |- H C_ H
102	101	a1i 11	. . . . . . . 8 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> H C_ H )
103	7, 8, 45, 46, 47, 49, 100, 102	ss2mcls 31465	. . . . . . 7 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> ( ( 1st ` ( 1st ` x ) ) (mCls ` T ) H ) C_ ( M (mCls ` T ) H ) )
104	89, 51	eqeltrrd 2702	. . . . . . . 8 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> <. ( 1st ` ( 1st ` x ) ) , H , A >. e. J )
105	3, 42, 45	elmpps 31470	. . . . . . . . 9 |- ( <. ( 1st ` ( 1st ` x ) ) , H , A >. e. J <-> ( <. ( 1st ` ( 1st ` x ) ) , H , A >. e. (mPreSt ` T ) /\ A e. ( ( 1st ` ( 1st ` x ) ) (mCls ` T ) H ) ) )
106	105	simprbi 480	. . . . . . . 8 |- ( <. ( 1st ` ( 1st ` x ) ) , H , A >. e. J -> A e. ( ( 1st ` ( 1st ` x ) ) (mCls ` T ) H ) )
107	104, 106	syl 17	. . . . . . 7 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> A e. ( ( 1st ` ( 1st ` x ) ) (mCls ` T ) H ) )
108	103, 107	sseldd 3604	. . . . . 6 |- ( ( ( T e. mFS /\ <. C , H , A >. e. U ) /\ ( x e. J /\ ( R ` x ) = ( R ` <. C , H , A >. ) ) ) -> A e. ( M (mCls ` T ) H ) )
109	44, 108	rexlimddv 3035	. . . . 5 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> A e. ( M (mCls ` T ) H ) )
110	3, 42, 45	elmpps 31470	. . . . 5 |- ( <. M , H , A >. e. J <-> ( <. M , H , A >. e. (mPreSt ` T ) /\ A e. ( M (mCls ` T ) H ) ) )
111	40, 109, 110	sylanbrc 698	. . . 4 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> <. M , H , A >. e. J )
112	1	ineq1i 3810	. . . . . . . 8 |- ( M i^i ( Z X. Z ) ) = ( ( C u. ( D \ ( Z X. Z ) ) ) i^i ( Z X. Z ) )
113		indir 3875	. . . . . . . 8 |- ( ( C u. ( D \ ( Z X. Z ) ) ) i^i ( Z X. Z ) ) = ( ( C i^i ( Z X. Z ) ) u. ( ( D \ ( Z X. Z ) ) i^i ( Z X. Z ) ) )
114		incom 3805	. . . . . . . . . . 11 |- ( ( D \ ( Z X. Z ) ) i^i ( Z X. Z ) ) = ( ( Z X. Z ) i^i ( D \ ( Z X. Z ) ) )
115		disjdif 4040	. . . . . . . . . . 11 |- ( ( Z X. Z ) i^i ( D \ ( Z X. Z ) ) ) = (/)
116	114, 115	eqtri 2644	. . . . . . . . . 10 |- ( ( D \ ( Z X. Z ) ) i^i ( Z X. Z ) ) = (/)
117		0ss 3972	. . . . . . . . . 10 |- (/) C_ ( C i^i ( Z X. Z ) )
118	116, 117	eqsstri 3635	. . . . . . . . 9 |- ( ( D \ ( Z X. Z ) ) i^i ( Z X. Z ) ) C_ ( C i^i ( Z X. Z ) )
119		ssequn2 3786	. . . . . . . . 9 |- ( ( ( D \ ( Z X. Z ) ) i^i ( Z X. Z ) ) C_ ( C i^i ( Z X. Z ) ) <-> ( ( C i^i ( Z X. Z ) ) u. ( ( D \ ( Z X. Z ) ) i^i ( Z X. Z ) ) ) = ( C i^i ( Z X. Z ) ) )
120	118, 119	mpbi 220	. . . . . . . 8 |- ( ( C i^i ( Z X. Z ) ) u. ( ( D \ ( Z X. Z ) ) i^i ( Z X. Z ) ) ) = ( C i^i ( Z X. Z ) )
121	112, 113, 120	3eqtri 2648	. . . . . . 7 |- ( M i^i ( Z X. Z ) ) = ( C i^i ( Z X. Z ) )
122	121	a1i 11	. . . . . 6 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> ( M i^i ( Z X. Z ) ) = ( C i^i ( Z X. Z ) ) )
123	122	oteq1d 4414	. . . . 5 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> <. ( M i^i ( Z X. Z ) ) , H , A >. = <. ( C i^i ( Z X. Z ) ) , H , A >. )
124	59, 3, 41, 63	msrval 31435	. . . . . 6 |- ( <. M , H , A >. e. (mPreSt ` T ) -> ( R ` <. M , H , A >. ) = <. ( M i^i ( Z X. Z ) ) , H , A >. )
125	40, 124	syl 17	. . . . 5 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> ( R ` <. M , H , A >. ) = <. ( M i^i ( Z X. Z ) ) , H , A >. )
126	123, 125, 65	3eqtr4d 2666	. . . 4 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> ( R ` <. M , H , A >. ) = ( R ` <. C , H , A >. ) )
127	111, 126	jca 554	. . 3 |- ( ( T e. mFS /\ <. C , H , A >. e. U ) -> ( <. M , H , A >. e. J /\ ( R ` <. M , H , A >. ) = ( R ` <. C , H , A >. ) ) )
128	127	ex 450	. 2 |- ( T e. mFS -> ( <. C , H , A >. e. U -> ( <. M , H , A >. e. J /\ ( R ` <. M , H , A >. ) = ( R ` <. C , H , A >. ) ) ) )
129	41, 42, 2	mthmi 31474	. 2 |- ( ( <. M , H , A >. e. J /\ ( R ` <. M , H , A >. ) = ( R ` <. C , H , A >. ) ) -> <. C , H , A >. e. U )
130	128, 129	impbid1 215	1 |- ( T e. mFS -> ( <. C , H , A >. e. U <-> ( <. M , H , A >. e. J /\ ( R ` <. M , H , A >. ) = ( R ` <. C , H , A >. ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   <.cotp 4185   U.cuni 4436   _I cid 5023   X. cxp 5112   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Fincfn 7955  mVRcmvar 31358  mExcmex 31364  mDVcmdv 31365  mVarscmvrs 31366  mPreStcmpst 31370  mStRedcmsr 31371  mFScmfs 31373  mClscmcls 31374  mPPStcmpps 31375  mThmcmthm 31376

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-frmd 17386  df-mrex 31383  df-mex 31384  df-mdv 31385  df-mrsub 31387  df-msub 31388  df-mvh 31389  df-mpst 31390  df-msr 31391  df-msta 31392  df-mfs 31393  df-mcls 31394  df-mpps 31395  df-mthm 31396

This theorem is referenced by: (None)