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Theorem msrf 29186
Description: The reduct of a pre-statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mpstssv.p  |-  P  =  (mPreSt `  T )
msrf.r  |-  R  =  (mStRed `  T )
Assertion
Ref Expression
msrf  |-  R : P
--> P

Proof of Theorem msrf
Dummy variables  h  a  s  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 otex 4721 . . . . 5  |-  <. (
( 1st `  ( 1st `  s ) )  i^i  [_ U. ( (mVars `  T ) " (
h  u.  { a } ) )  / 
z ]_ ( z  X.  z ) ) ,  h ,  a >.  e.  _V
21csbex 4590 . . . 4  |-  [_ ( 2nd `  s )  / 
a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( (mVars `  T ) " ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >.  e.  _V
32csbex 4590 . . 3  |-  [_ ( 2nd `  ( 1st `  s
) )  /  h ]_ [_ ( 2nd `  s
)  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( (mVars `  T ) " (
h  u.  { a } ) )  / 
z ]_ ( z  X.  z ) ) ,  h ,  a >.  e.  _V
4 eqid 2457 . . . 4  |-  (mVars `  T )  =  (mVars `  T )
5 mpstssv.p . . . 4  |-  P  =  (mPreSt `  T )
6 msrf.r . . . 4  |-  R  =  (mStRed `  T )
74, 5, 6msrfval 29181 . . 3  |-  R  =  ( s  e.  P  |-> 
[_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  / 
a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( (mVars `  T ) " ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
83, 7fnmpti 5715 . 2  |-  R  Fn  P
95mpst123 29184 . . . . . 6  |-  ( s  e.  P  ->  s  =  <. ( 1st `  ( 1st `  s ) ) ,  ( 2nd `  ( 1st `  s ) ) ,  ( 2nd `  s
) >. )
109fveq2d 5876 . . . . 5  |-  ( s  e.  P  ->  ( R `  s )  =  ( R `  <. ( 1st `  ( 1st `  s ) ) ,  ( 2nd `  ( 1st `  s ) ) ,  ( 2nd `  s
) >. ) )
11 id 22 . . . . . . 7  |-  ( s  e.  P  ->  s  e.  P )
129, 11eqeltrrd 2546 . . . . . 6  |-  ( s  e.  P  ->  <. ( 1st `  ( 1st `  s
) ) ,  ( 2nd `  ( 1st `  s ) ) ,  ( 2nd `  s
) >.  e.  P )
13 eqid 2457 . . . . . . 7  |-  U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  =  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) )
144, 5, 6, 13msrval 29182 . . . . . 6  |-  ( <.
( 1st `  ( 1st `  s ) ) ,  ( 2nd `  ( 1st `  s ) ) ,  ( 2nd `  s
) >.  e.  P  -> 
( R `  <. ( 1st `  ( 1st `  s ) ) ,  ( 2nd `  ( 1st `  s ) ) ,  ( 2nd `  s
) >. )  =  <. ( ( 1st `  ( 1st `  s ) )  i^i  ( U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) ) ,  ( 2nd `  ( 1st `  s ) ) ,  ( 2nd `  s
) >. )
1512, 14syl 16 . . . . 5  |-  ( s  e.  P  ->  ( R `  <. ( 1st `  ( 1st `  s
) ) ,  ( 2nd `  ( 1st `  s ) ) ,  ( 2nd `  s
) >. )  =  <. ( ( 1st `  ( 1st `  s ) )  i^i  ( U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) ) ,  ( 2nd `  ( 1st `  s ) ) ,  ( 2nd `  s
) >. )
1610, 15eqtrd 2498 . . . 4  |-  ( s  e.  P  ->  ( R `  s )  =  <. ( ( 1st `  ( 1st `  s
) )  i^i  ( U. ( (mVars `  T
) " ( ( 2nd `  ( 1st `  s ) )  u. 
{ ( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) ) ,  ( 2nd `  ( 1st `  s ) ) ,  ( 2nd `  s
) >. )
17 inss1 3714 . . . . . . 7  |-  ( ( 1st `  ( 1st `  s ) )  i^i  ( U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) )  X.  U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) ) ) )  C_  ( 1st `  ( 1st `  s
) )
18 eqid 2457 . . . . . . . . . . 11  |-  (mDV `  T )  =  (mDV
`  T )
19 eqid 2457 . . . . . . . . . . 11  |-  (mEx `  T )  =  (mEx
`  T )
2018, 19, 5elmpst 29180 . . . . . . . . . 10  |-  ( <.
( 1st `  ( 1st `  s ) ) ,  ( 2nd `  ( 1st `  s ) ) ,  ( 2nd `  s
) >.  e.  P  <->  ( (
( 1st `  ( 1st `  s ) ) 
C_  (mDV `  T
)  /\  `' ( 1st `  ( 1st `  s
) )  =  ( 1st `  ( 1st `  s ) ) )  /\  ( ( 2nd `  ( 1st `  s
) )  C_  (mEx `  T )  /\  ( 2nd `  ( 1st `  s
) )  e.  Fin )  /\  ( 2nd `  s
)  e.  (mEx `  T ) ) )
2112, 20sylib 196 . . . . . . . . 9  |-  ( s  e.  P  ->  (
( ( 1st `  ( 1st `  s ) ) 
C_  (mDV `  T
)  /\  `' ( 1st `  ( 1st `  s
) )  =  ( 1st `  ( 1st `  s ) ) )  /\  ( ( 2nd `  ( 1st `  s
) )  C_  (mEx `  T )  /\  ( 2nd `  ( 1st `  s
) )  e.  Fin )  /\  ( 2nd `  s
)  e.  (mEx `  T ) ) )
2221simp1d 1008 . . . . . . . 8  |-  ( s  e.  P  ->  (
( 1st `  ( 1st `  s ) ) 
C_  (mDV `  T
)  /\  `' ( 1st `  ( 1st `  s
) )  =  ( 1st `  ( 1st `  s ) ) ) )
2322simpld 459 . . . . . . 7  |-  ( s  e.  P  ->  ( 1st `  ( 1st `  s
) )  C_  (mDV `  T ) )
2417, 23syl5ss 3510 . . . . . 6  |-  ( s  e.  P  ->  (
( 1st `  ( 1st `  s ) )  i^i  ( U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) )  C_  (mDV `  T ) )
25 cnvin 5420 . . . . . . 7  |-  `' ( ( 1st `  ( 1st `  s ) )  i^i  ( U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) )  =  ( `' ( 1st `  ( 1st `  s
) )  i^i  `' ( U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) )  X.  U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) ) ) )
2622simprd 463 . . . . . . . 8  |-  ( s  e.  P  ->  `' ( 1st `  ( 1st `  s ) )  =  ( 1st `  ( 1st `  s ) ) )
27 cnvxp 5431 . . . . . . . . 9  |-  `' ( U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) )  X.  U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) ) )  =  ( U. ( (mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) )
2827a1i 11 . . . . . . . 8  |-  ( s  e.  P  ->  `' ( U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) )  X.  U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) ) )  =  ( U. ( (mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) )
2926, 28ineq12d 3697 . . . . . . 7  |-  ( s  e.  P  ->  ( `' ( 1st `  ( 1st `  s ) )  i^i  `' ( U. ( (mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) )  =  ( ( 1st `  ( 1st `  s ) )  i^i  ( U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) ) )
3025, 29syl5eq 2510 . . . . . 6  |-  ( s  e.  P  ->  `' ( ( 1st `  ( 1st `  s ) )  i^i  ( U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) )  =  ( ( 1st `  ( 1st `  s ) )  i^i  ( U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) ) )
3124, 30jca 532 . . . . 5  |-  ( s  e.  P  ->  (
( ( 1st `  ( 1st `  s ) )  i^i  ( U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) )  C_  (mDV `  T )  /\  `' ( ( 1st `  ( 1st `  s
) )  i^i  ( U. ( (mVars `  T
) " ( ( 2nd `  ( 1st `  s ) )  u. 
{ ( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) )  =  ( ( 1st `  ( 1st `  s ) )  i^i  ( U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) ) ) )
3221simp2d 1009 . . . . 5  |-  ( s  e.  P  ->  (
( 2nd `  ( 1st `  s ) ) 
C_  (mEx `  T
)  /\  ( 2nd `  ( 1st `  s
) )  e.  Fin ) )
3321simp3d 1010 . . . . 5  |-  ( s  e.  P  ->  ( 2nd `  s )  e.  (mEx `  T )
)
3418, 19, 5elmpst 29180 . . . . 5  |-  ( <.
( ( 1st `  ( 1st `  s ) )  i^i  ( U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) ) ,  ( 2nd `  ( 1st `  s ) ) ,  ( 2nd `  s
) >.  e.  P  <->  ( (
( ( 1st `  ( 1st `  s ) )  i^i  ( U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) )  C_  (mDV `  T )  /\  `' ( ( 1st `  ( 1st `  s
) )  i^i  ( U. ( (mVars `  T
) " ( ( 2nd `  ( 1st `  s ) )  u. 
{ ( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) )  =  ( ( 1st `  ( 1st `  s ) )  i^i  ( U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) ) )  /\  ( ( 2nd `  ( 1st `  s
) )  C_  (mEx `  T )  /\  ( 2nd `  ( 1st `  s
) )  e.  Fin )  /\  ( 2nd `  s
)  e.  (mEx `  T ) ) )
3531, 32, 33, 34syl3anbrc 1180 . . . 4  |-  ( s  e.  P  ->  <. (
( 1st `  ( 1st `  s ) )  i^i  ( U. (
(mVars `  T ) " ( ( 2nd `  ( 1st `  s
) )  u.  {
( 2nd `  s
) } ) )  X.  U. ( (mVars `  T ) " (
( 2nd `  ( 1st `  s ) )  u.  { ( 2nd `  s ) } ) ) ) ) ,  ( 2nd `  ( 1st `  s ) ) ,  ( 2nd `  s
) >.  e.  P )
3616, 35eqeltrd 2545 . . 3  |-  ( s  e.  P  ->  ( R `  s )  e.  P )
3736rgen 2817 . 2  |-  A. s  e.  P  ( R `  s )  e.  P
38 ffnfv 6058 . 2  |-  ( R : P --> P  <->  ( R  Fn  P  /\  A. s  e.  P  ( R `  s )  e.  P
) )
398, 37, 38mpbir2an 920 1  |-  R : P
--> P
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   [_csb 3430    u. cun 3469    i^i cin 3470    C_ wss 3471   {csn 4032   <.cotp 4040   U.cuni 4251    X. cxp 5006   `'ccnv 5007   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594   1stc1st 6797   2ndc2nd 6798   Fincfn 7535  mExcmex 29111  mDVcmdv 29112  mVarscmvrs 29113  mPreStcmpst 29117  mStRedcmsr 29118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-ot 4041  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-1st 6799  df-2nd 6800  df-mpst 29137  df-msr 29138
This theorem is referenced by:  msrrcl  29187  msrid  29189  msrfo  29190  mstapst  29191  elmsta  29192  elmthm  29220  mthmsta  29222  mthmblem  29224
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