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Theorem mppsval 28798
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 5852 . . . . . . . 8  |-  ( t  =  T  ->  (mPreSt `  t )  =  (mPreSt `  T ) )
3 mppsval.p . . . . . . . 8  |-  P  =  (mPreSt `  T )
42, 3syl6eqr 2500 . . . . . . 7  |-  ( t  =  T  ->  (mPreSt `  t )  =  P )
54eleq2d 2511 . . . . . 6  |-  ( t  =  T  ->  ( <. d ,  h ,  a >.  e.  (mPreSt `  t )  <->  <. d ,  h ,  a >.  e.  P ) )
6 fveq2 5852 . . . . . . . . 9  |-  ( t  =  T  ->  (mCls `  t )  =  (mCls `  T ) )
7 mppsval.c . . . . . . . . 9  |-  C  =  (mCls `  T )
86, 7syl6eqr 2500 . . . . . . . 8  |-  ( t  =  T  ->  (mCls `  t )  =  C )
98oveqd 6294 . . . . . . 7  |-  ( t  =  T  ->  (
d (mCls `  t
) h )  =  ( d C h ) )
109eleq2d 2511 . . . . . 6  |-  ( t  =  T  ->  (
a  e.  ( d (mCls `  t )
h )  <->  a  e.  ( d C h ) ) )
115, 10anbi12d 710 . . . . 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 6332 . . . 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 28724 . . . 4  |- mPPSt  =  ( t  e.  _V  |->  {
<. <. d ,  h >. ,  a >.  |  (
<. d ,  h ,  a >.  e.  (mPreSt `  t )  /\  a  e.  ( d (mCls `  t ) h ) ) } )
14 fvex 5862 . . . . . 6  |-  (mPreSt `  T )  e.  _V
153, 14eqeltri 2525 . . . . 5  |-  P  e. 
_V
163, 1, 7mppspstlem 28797 . . . . 5  |-  { <. <.
d ,  h >. ,  a >.  |  ( <. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) }  C_  P
1715, 16ssexi 4578 . . . 4  |-  { <. <.
d ,  h >. ,  a >.  |  ( <. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) }  e.  _V
1812, 13, 17fvmpt 5937 . . 3  |-  ( T  e.  _V  ->  (mPPSt `  T )  =  { <. <. d ,  h >. ,  a >.  |  (
<. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) } )
19 fvprc 5846 . . . 4  |-  ( -.  T  e.  _V  ->  (mPPSt `  T )  =  (/) )
20 df-oprab 6281 . . . . 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 3786 . . . . . . 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 5879 . . . . . . . . . . 11  |-  ( <.
d ,  h ,  a >.  e.  (mPreSt `  T )  ->  T  e.  _V )
2322, 3eleq2s 2549 . . . . . . . . . 10  |-  ( <.
d ,  h ,  a >.  e.  P  ->  T  e.  _V )
2423ad2antrl 727 . . . . . . . . 9  |-  ( ( x  =  <. <. d ,  h >. ,  a >.  /\  ( <. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) )  ->  T  e.  _V )
2524exlimivv 1708 . . . . . . . 8  |-  ( E. h E. a ( x  =  <. <. d ,  h >. ,  a >.  /\  ( <. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) )  ->  T  e.  _V )
2625exlimivv 1708 . . . . . . 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 195 . . . . . 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 2674 . . . . 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 2494 . . . 4  |-  ( -.  T  e.  _V  ->  {
<. <. d ,  h >. ,  a >.  |  (
<. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) }  =  (/) )
3019, 29eqtr4d 2485 . . 3  |-  ( -.  T  e.  _V  ->  (mPPSt `  T )  =  { <. <. d ,  h >. ,  a >.  |  (
<. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) } )
3118, 30pm2.61i 164 . 2  |-  (mPPSt `  T )  =  { <. <. d ,  h >. ,  a >.  |  (
<. d ,  h ,  a >.  e.  P  /\  a  e.  (
d C h ) ) }
321, 31eqtri 2470 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 369    = wceq 1381   E.wex 1597    e. wcel 1802   {cab 2426    =/= wne 2636   _Vcvv 3093   (/)c0 3767   <.cop 4016   <.cotp 4018   ` cfv 5574  (class class class)co 6277   {coprab 6278  mPreStcmpst 28699  mClscmcls 28703  mPPStcmpps 28704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-ot 4019  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-iota 5537  df-fun 5576  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpps 28724
This theorem is referenced by:  elmpps  28799  mppspst  28800
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