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Theorem mppsval 30282
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.
StepHypRef Expression
1 mppsval.j . 2  |-  J  =  (mPPSt `  T )
2 fveq2 5879 . . . . . . . 8  |-  ( t  =  T  ->  (mPreSt `  t )  =  (mPreSt `  T ) )
3 mppsval.p . . . . . . . 8  |-  P  =  (mPreSt `  T )
42, 3syl6eqr 2523 . . . . . . 7  |-  ( t  =  T  ->  (mPreSt `  t )  =  P )
54eleq2d 2534 . . . . . 6  |-  ( t  =  T  ->  ( <. d ,  h ,  a >.  e.  (mPreSt `  t )  <->  <. d ,  h ,  a >.  e.  P ) )
6 fveq2 5879 . . . . . . . . 9  |-  ( t  =  T  ->  (mCls `  t )  =  (mCls `  T ) )
7 mppsval.c . . . . . . . . 9  |-  C  =  (mCls `  T )
86, 7syl6eqr 2523 . . . . . . . 8  |-  ( t  =  T  ->  (mCls `  t )  =  C )
98oveqd 6325 . . . . . . 7  |-  ( t  =  T  ->  (
d (mCls `  t
) h )  =  ( d C h ) )
109eleq2d 2534 . . . . . 6  |-  ( t  =  T  ->  (
a  e.  ( d (mCls `  t )
h )  <->  a  e.  ( d C h ) ) )
115, 10anbi12d 725 . . . . 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 ) ) ) )
1211oprabbidv 6364 . . . 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 30208 . . . 4  |- mPPSt  =  ( t  e.  _V  |->  {
<. <. d ,  h >. ,  a >.  |  (
<. d ,  h ,  a >.  e.  (mPreSt `  t )  /\  a  e.  ( d (mCls `  t ) h ) ) } )
14 fvex 5889 . . . . . 6  |-  (mPreSt `  T )  e.  _V
153, 14eqeltri 2545 . . . . 5  |-  P  e. 
_V
163, 1, 7mppspstlem 30281 . . . . 5  |-  { <. <.
d ,  h >. ,  a >.  |  ( <. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) }  C_  P
1715, 16ssexi 4541 . . . 4  |-  { <. <.
d ,  h >. ,  a >.  |  ( <. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) }  e.  _V
1812, 13, 17fvmpt 5963 . . 3  |-  ( T  e.  _V  ->  (mPPSt `  T )  =  { <. <. d ,  h >. ,  a >.  |  (
<. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) } )
19 fvprc 5873 . . . 4  |-  ( -.  T  e.  _V  ->  (mPPSt `  T )  =  (/) )
20 df-oprab 6312 . . . . 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 3754 . . . . . . 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 5906 . . . . . . . . . . 11  |-  ( <.
d ,  h ,  a >.  e.  (mPreSt `  T )  ->  T  e.  _V )
2322, 3eleq2s 2567 . . . . . . . . . 10  |-  ( <.
d ,  h ,  a >.  e.  P  ->  T  e.  _V )
2423ad2antrl 742 . . . . . . . . 9  |-  ( ( x  =  <. <. d ,  h >. ,  a >.  /\  ( <. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) )  ->  T  e.  _V )
2524exlimivv 1786 . . . . . . . 8  |-  ( E. h E. a ( x  =  <. <. d ,  h >. ,  a >.  /\  ( <. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) )  ->  T  e.  _V )
2625exlimivv 1786 . . . . . . 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 )
2721, 26sylbi 200 . . . . . 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 )
2827necon1bi 2671 . . . . 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 ) ) ) }  =  (/) )
2920, 28syl5eq 2517 . . . 4  |-  ( -.  T  e.  _V  ->  {
<. <. d ,  h >. ,  a >.  |  (
<. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) }  =  (/) )
3019, 29eqtr4d 2508 . . 3  |-  ( -.  T  e.  _V  ->  (mPPSt `  T )  =  { <. <. d ,  h >. ,  a >.  |  (
<. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) } )
3118, 30pm2.61i 169 . 2  |-  (mPPSt `  T )  =  { <. <. d ,  h >. ,  a >.  |  (
<. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) }
321, 31eqtri 2493 1  |-  J  =  { <. <. d ,  h >. ,  a >.  |  (
<. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457    =/= wne 2641   _Vcvv 3031   (/)c0 3722   <.cop 3965   <.cotp 3967   ` cfv 5589  (class class class)co 6308   {coprab 6309  mPreStcmpst 30183  mClscmcls 30187  mPPStcmpps 30188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-ot 3968  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpps 30208
This theorem is referenced by:  elmpps  30283  mppspst  30284
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