Theorem mppsval 31469

Description: Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mppsval.p	|- P = (mPreSt ` T )
mppsval.j	|- J = (mPPSt ` T )
mppsval.c	|- C = (mCls ` T )

Assertion

Ref	Expression
mppsval	|- J = { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) }

Distinct variable groups:   a, d, h, C   P, a, d, h   T, a, d, h

Allowed substitution hints:   J( h, a, d)

Proof of Theorem mppsval

Dummy variables t x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mppsval.j	. 2 |- J = (mPPSt ` T )
2		fveq2 6191	. . . . . . . 8 |- ( t = T -> (mPreSt ` t ) = (mPreSt ` T ) )
3		mppsval.p	. . . . . . . 8 |- P = (mPreSt ` T )
4	2, 3	syl6eqr 2674	. . . . . . 7 |- ( t = T -> (mPreSt ` t ) = P )
5	4	eleq2d 2687	. . . . . 6 |- ( t = T -> ( <. d , h , a >. e. (mPreSt ` t ) <-> <. d , h , a >. e. P ) )
6		fveq2 6191	. . . . . . . . 9 |- ( t = T -> (mCls ` t ) = (mCls ` T ) )
7		mppsval.c	. . . . . . . . 9 |- C = (mCls ` T )
8	6, 7	syl6eqr 2674	. . . . . . . 8 |- ( t = T -> (mCls ` t ) = C )
9	8	oveqd 6667	. . . . . . 7 |- ( t = T -> ( d (mCls ` t ) h ) = ( d C h ) )
10	9	eleq2d 2687	. . . . . 6 |- ( t = T -> ( a e. ( d (mCls ` t ) h ) <-> a e. ( d C h ) ) )
11	5, 10	anbi12d 747	. . . . 5 |- ( t = T -> ( ( <. d , h , a >. e. (mPreSt ` t ) /\ a e. ( d (mCls ` t ) h ) ) <-> ( <. d , h , a >. e. P /\ a e. ( d C h ) ) ) )
12	11	oprabbidv 6709	. . . 4 |- ( t = T -> { <. <. d , h >. , a >. | ( <. d , h , a >. e. (mPreSt ` t ) /\ a e. ( d (mCls ` t ) h ) ) } = { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } )
13		df-mpps 31395	. . . 4 |- mPPSt = ( t e. _V |-> { <. <. d , h >. , a >. | ( <. d , h , a >. e. (mPreSt ` t ) /\ a e. ( d (mCls ` t ) h ) ) } )
14		fvex 6201	. . . . . 6 |- (mPreSt ` T ) e. _V
15	3, 14	eqeltri 2697	. . . . 5 |- P e. _V
16	3, 1, 7	mppspstlem 31468	. . . . 5 |- { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } C_ P
17	15, 16	ssexi 4803	. . . 4 |- { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } e. _V
18	12, 13, 17	fvmpt 6282	. . 3 |- ( T e. _V -> (mPPSt ` T ) = { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } )
19		fvprc 6185	. . . 4 |- ( -. T e. _V -> (mPPSt ` T ) = (/) )
20		df-oprab 6654	. . . . 5 |- { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } = { x | E. d E. h E. a ( x = <. <. d , h >. , a >. /\ ( <. d , h , a >. e. P /\ a e. ( d C h ) ) ) }
21		abn0 3954	. . . . . . 7 |- ( { x | E. d E. h E. a ( x = <. <. d , h >. , a >. /\ ( <. d , h , a >. e. P /\ a e. ( d C h ) ) ) } =/= (/) <-> E. x E. d E. h E. a ( x = <. <. d , h >. , a >. /\ ( <. d , h , a >. e. P /\ a e. ( d C h ) ) ) )
22		elfvex 6221	. . . . . . . . . . 11 |- ( <. d , h , a >. e. (mPreSt ` T ) -> T e. _V )
23	22, 3	eleq2s 2719	. . . . . . . . . 10 |- ( <. d , h , a >. e. P -> T e. _V )
24	23	ad2antrl 764	. . . . . . . . 9 |- ( ( x = <. <. d , h >. , a >. /\ ( <. d , h , a >. e. P /\ a e. ( d C h ) ) ) -> T e. _V )
25	24	exlimivv 1860	. . . . . . . 8 |- ( E. h E. a ( x = <. <. d , h >. , a >. /\ ( <. d , h , a >. e. P /\ a e. ( d C h ) ) ) -> T e. _V )
26	25	exlimivv 1860	. . . . . . 7 |- ( E. x E. d E. h E. a ( x = <. <. d , h >. , a >. /\ ( <. d , h , a >. e. P /\ a e. ( d C h ) ) ) -> T e. _V )
27	21, 26	sylbi 207	. . . . . 6 |- ( { x | E. d E. h E. a ( x = <. <. d , h >. , a >. /\ ( <. d , h , a >. e. P /\ a e. ( d C h ) ) ) } =/= (/) -> T e. _V )
28	27	necon1bi 2822	. . . . 5 |- ( -. T e. _V -> { x | E. d E. h E. a ( x = <. <. d , h >. , a >. /\ ( <. d , h , a >. e. P /\ a e. ( d C h ) ) ) } = (/) )
29	20, 28	syl5eq 2668	. . . 4 |- ( -. T e. _V -> { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } = (/) )
30	19, 29	eqtr4d 2659	. . 3 |- ( -. T e. _V -> (mPPSt ` T ) = { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } )
31	18, 30	pm2.61i 176	. 2 |- (mPPSt ` T ) = { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) }
32	1, 31	eqtri 2644	1 |- J = { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) }

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   _Vcvv 3200   (/)c0 3915   <.cop 4183   <.cotp 4185   ` cfv 5888  (class class class)co 6650   {coprab 6651  mPreStcmpst 31370  mClscmcls 31374  mPPStcmpps 31375

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-uni 4437  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpps 31395

This theorem is referenced by:  elmpps  31470  mppspst  31471