Theorem mppspstlem 31468

Description: Lemma for mppspst 31471. (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
mppspstlem	|- { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } C_ P

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 mppspstlem

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oprab 6654	. 2 |- { <. <. 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 ) ) ) }
2		df-ot 4186	. . . . . . . . . 10 |- <. d , h , a >. = <. <. d , h >. , a >.
3	2	eqeq2i 2634	. . . . . . . . 9 |- ( x = <. d , h , a >. <-> x = <. <. d , h >. , a >. )
4	3	biimpri 218	. . . . . . . 8 |- ( x = <. <. d , h >. , a >. -> x = <. d , h , a >. )
5	4	eleq1d 2686	. . . . . . 7 |- ( x = <. <. d , h >. , a >. -> ( x e. P <-> <. d , h , a >. e. P ) )
6	5	biimpar 502	. . . . . 6 |- ( ( x = <. <. d , h >. , a >. /\ <. d , h , a >. e. P ) -> x e. P )
7	6	adantrr 753	. . . . 5 |- ( ( x = <. <. d , h >. , a >. /\ ( <. d , h , a >. e. P /\ a e. ( d C h ) ) ) -> x e. P )
8	7	exlimiv 1858	. . . 4 |- ( E. a ( x = <. <. d , h >. , a >. /\ ( <. d , h , a >. e. P /\ a e. ( d C h ) ) ) -> x e. P )
9	8	exlimivv 1860	. . 3 |- ( E. d E. h E. a ( x = <. <. d , h >. , a >. /\ ( <. d , h , a >. e. P /\ a e. ( d C h ) ) ) -> x e. P )
10	9	abssi 3677	. 2 |- { x | E. d E. h E. a ( x = <. <. d , h >. , a >. /\ ( <. d , h , a >. e. P /\ a e. ( d C h ) ) ) } C_ P
11	1, 10	eqsstri 3635	1 |- { <. <. d , h >. , a >. | ( <. d , h , a >. e. P /\ a e. ( d C h ) ) } C_ P

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   C_ wss 3574   <.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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-in 3581  df-ss 3588  df-ot 4186  df-oprab 6654

This theorem is referenced by:  mppsval  31469  mppspst  31471