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Theorem msrrcl 30231
Description: If  X and  Y have the same reduct, then one is a pre-statement iff the other is. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mpstssv.p  |-  P  =  (mPreSt `  T )
msrf.r  |-  R  =  (mStRed `  T )
Assertion
Ref Expression
msrrcl  |-  ( ( R `  X )  =  ( R `  Y )  ->  ( X  e.  P  <->  Y  e.  P ) )

Proof of Theorem msrrcl
StepHypRef Expression
1 mpstssv.p . . . . 5  |-  P  =  (mPreSt `  T )
2 msrf.r . . . . 5  |-  R  =  (mStRed `  T )
31, 2msrf 30230 . . . 4  |-  R : P
--> P
43ffvelrni 6049 . . 3  |-  ( X  e.  P  ->  ( R `  X )  e.  P )
54a1i 11 . 2  |-  ( ( R `  X )  =  ( R `  Y )  ->  ( X  e.  P  ->  ( R `  X )  e.  P ) )
63ffvelrni 6049 . . 3  |-  ( Y  e.  P  ->  ( R `  Y )  e.  P )
7 eleq1 2528 . . 3  |-  ( ( R `  X )  =  ( R `  Y )  ->  (
( R `  X
)  e.  P  <->  ( R `  Y )  e.  P
) )
86, 7syl5ibr 229 . 2  |-  ( ( R `  X )  =  ( R `  Y )  ->  ( Y  e.  P  ->  ( R `  X )  e.  P ) )
93fdmi 5761 . . . . . 6  |-  dom  R  =  P
10 0nelxp 4884 . . . . . . 7  |-  -.  (/)  e.  ( ( _V  X.  _V )  X.  _V )
111mpstssv 30227 . . . . . . . 8  |-  P  C_  ( ( _V  X.  _V )  X.  _V )
1211sseli 3440 . . . . . . 7  |-  ( (/)  e.  P  ->  (/)  e.  ( ( _V  X.  _V )  X.  _V ) )
1310, 12mto 181 . . . . . 6  |-  -.  (/)  e.  P
149, 13ndmfvrcl 5917 . . . . 5  |-  ( ( R `  X )  e.  P  ->  X  e.  P )
1514adantl 472 . . . 4  |-  ( ( ( R `  X
)  =  ( R `
 Y )  /\  ( R `  X )  e.  P )  ->  X  e.  P )
167biimpa 491 . . . . 5  |-  ( ( ( R `  X
)  =  ( R `
 Y )  /\  ( R `  X )  e.  P )  -> 
( R `  Y
)  e.  P )
179, 13ndmfvrcl 5917 . . . . 5  |-  ( ( R `  Y )  e.  P  ->  Y  e.  P )
1816, 17syl 17 . . . 4  |-  ( ( ( R `  X
)  =  ( R `
 Y )  /\  ( R `  X )  e.  P )  ->  Y  e.  P )
1915, 182thd 248 . . 3  |-  ( ( ( R `  X
)  =  ( R `
 Y )  /\  ( R `  X )  e.  P )  -> 
( X  e.  P  <->  Y  e.  P ) )
2019ex 440 . 2  |-  ( ( R `  X )  =  ( R `  Y )  ->  (
( R `  X
)  e.  P  -> 
( X  e.  P  <->  Y  e.  P ) ) )
215, 8, 20pm5.21ndd 360 1  |-  ( ( R `  X )  =  ( R `  Y )  ->  ( X  e.  P  <->  Y  e.  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   _Vcvv 3057   (/)c0 3743    X. cxp 4854   ` cfv 5605  mPreStcmpst 30161  mStRedcmsr 30162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-ot 3989  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-1st 6825  df-2nd 6826  df-mpst 30181  df-msr 30182
This theorem is referenced by:  elmthm  30264
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