Theorem elmpps 31470

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
elmpps	|- ( <. D , H , A >. e. J <-> ( <. D , H , A >. e. P /\ A e. ( D C H ) ) )

Proof of Theorem elmpps

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

Step	Hyp	Ref	Expression
1		df-ot 4186	. . 3 |- <. D , H , A >. = <. <. D , H >. , A >.
2		mppsval.p	. . . 4 |- P = (mPreSt ` T )
3		mppsval.j	. . . 4 |- J = (mPPSt ` T )
4		mppsval.c	. . . 4 |- C = (mCls ` T )
5	2, 3, 4	mppsval 31469	. . 3 |- J = { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) }
6	1, 5	eleq12i 2694	. 2 |- ( <. D , H , A >. e. J <-> <. <. D , H >. , A >. e. { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } )
7		oprabss 6746	. . . 4 |- { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } C_ ( ( _V X. _V ) X. _V )
8	7	sseli 3599	. . 3 |- ( <. <. D , H >. , A >. e. { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } -> <. <. D , H >. , A >. e. ( ( _V X. _V ) X. _V ) )
9	2	mpstssv 31436	. . . . . 6 |- P C_ ( ( _V X. _V ) X. _V )
10	9	sseli 3599	. . . . 5 |- ( <. D , H , A >. e. P -> <. D , H , A >. e. ( ( _V X. _V ) X. _V ) )
11	1, 10	syl5eqelr 2706	. . . 4 |- ( <. D , H , A >. e. P -> <. <. D , H >. , A >. e. ( ( _V X. _V ) X. _V ) )
12	11	adantr 481	. . 3 |- ( ( <. D , H , A >. e. P /\ A e. ( D C H ) ) -> <. <. D , H >. , A >. e. ( ( _V X. _V ) X. _V ) )
13		opelxp 5146	. . . 4 |- ( <. <. D , H >. , A >. e. ( ( _V X. _V ) X. _V ) <-> ( <. D , H >. e. ( _V X. _V ) /\ A e. _V ) )
14		opelxp 5146	. . . . 5 |- ( <. D , H >. e. ( _V X. _V ) <-> ( D e. _V /\ H e. _V ) )
15		simp1 1061	. . . . . . . . . 10 |- ( ( d = D /\ h = H /\ a = A ) -> d = D )
16		simp2 1062	. . . . . . . . . 10 |- ( ( d = D /\ h = H /\ a = A ) -> h = H )
17		simp3 1063	. . . . . . . . . 10 |- ( ( d = D /\ h = H /\ a = A ) -> a = A )
18	15, 16, 17	oteq123d 4417	. . . . . . . . 9 |- ( ( d = D /\ h = H /\ a = A ) -> <. d , h , a >. = <. D , H , A >. )
19	18	eleq1d 2686	. . . . . . . 8 |- ( ( d = D /\ h = H /\ a = A ) -> ( <. d , h , a >. e. P <-> <. D , H , A >. e. P ) )
20	15, 16	oveq12d 6668	. . . . . . . . 9 |- ( ( d = D /\ h = H /\ a = A ) -> ( d C h ) = ( D C H ) )
21	17, 20	eleq12d 2695	. . . . . . . 8 |- ( ( d = D /\ h = H /\ a = A ) -> ( a e. ( d C h ) <-> A e. ( D C H ) ) )
22	19, 21	anbi12d 747	. . . . . . 7 |- ( ( d = D /\ h = H /\ a = A ) -> ( ( <. d , h , a >. e. P /\ a e. ( d C h ) ) <-> ( <. D , H , A >. e. P /\ A e. ( D C H ) ) ) )
23	22	eloprabga 6747	. . . . . 6 |- ( ( D e. _V /\ H e. _V /\ A e. _V ) -> ( <. <. D , H >. , A >. e. { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } <-> ( <. D , H , A >. e. P /\ A e. ( D C H ) ) ) )
24	23	3expa 1265	. . . . 5 |- ( ( ( D e. _V /\ H e. _V ) /\ A e. _V ) -> ( <. <. D , H >. , A >. e. { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } <-> ( <. D , H , A >. e. P /\ A e. ( D C H ) ) ) )
25	14, 24	sylanb 489	. . . 4 |- ( ( <. D , H >. e. ( _V X. _V ) /\ A e. _V ) -> ( <. <. D , H >. , A >. e. { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } <-> ( <. D , H , A >. e. P /\ A e. ( D C H ) ) ) )
26	13, 25	sylbi 207	. . 3 |- ( <. <. D , H >. , A >. e. ( ( _V X. _V ) X. _V ) -> ( <. <. D , H >. , A >. e. { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } <-> ( <. D , H , A >. e. P /\ A e. ( D C H ) ) ) )
27	8, 12, 26	pm5.21nii 368	. 2 |- ( <. <. D , H >. , A >. e. { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } <-> ( <. D , H , A >. e. P /\ A e. ( D C H ) ) )
28	6, 27	bitri 264	1 |- ( <. D , H , A >. e. J <-> ( <. D , H , A >. e. P /\ A e. ( D C H ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   <.cotp 4185   X. cxp 5112   ` 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  ax-un 6949

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-pw 4160  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-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpst 31390  df-mpps 31395

This theorem is referenced by:  mthmpps  31479  mclspps  31481