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Theorem mstaval 29970
Description: Value of the set of statements. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mstaval.r  |-  R  =  (mStRed `  T )
mstaval.s  |-  S  =  (mStat `  T )
Assertion
Ref Expression
mstaval  |-  S  =  ran  R

Proof of Theorem mstaval
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 mstaval.s . 2  |-  S  =  (mStat `  T )
2 fveq2 5881 . . . . . 6  |-  ( t  =  T  ->  (mStRed `  t )  =  (mStRed `  T ) )
3 mstaval.r . . . . . 6  |-  R  =  (mStRed `  T )
42, 3syl6eqr 2488 . . . . 5  |-  ( t  =  T  ->  (mStRed `  t )  =  R )
54rneqd 5082 . . . 4  |-  ( t  =  T  ->  ran  (mStRed `  t )  =  ran  R )
6 df-msta 29921 . . . 4  |- mStat  =  ( t  e.  _V  |->  ran  (mStRed `  t )
)
7 fvex 5891 . . . . . 6  |-  (mStRed `  T )  e.  _V
83, 7eqeltri 2513 . . . . 5  |-  R  e. 
_V
98rnex 6741 . . . 4  |-  ran  R  e.  _V
105, 6, 9fvmpt 5964 . . 3  |-  ( T  e.  _V  ->  (mStat `  T )  =  ran  R )
11 rn0 5106 . . . . 5  |-  ran  (/)  =  (/)
1211eqcomi 2442 . . . 4  |-  (/)  =  ran  (/)
13 fvprc 5875 . . . 4  |-  ( -.  T  e.  _V  ->  (mStat `  T )  =  (/) )
14 fvprc 5875 . . . . . 6  |-  ( -.  T  e.  _V  ->  (mStRed `  T )  =  (/) )
153, 14syl5eq 2482 . . . . 5  |-  ( -.  T  e.  _V  ->  R  =  (/) )
1615rneqd 5082 . . . 4  |-  ( -.  T  e.  _V  ->  ran 
R  =  ran  (/) )
1712, 13, 163eqtr4a 2496 . . 3  |-  ( -.  T  e.  _V  ->  (mStat `  T )  =  ran  R )
1810, 17pm2.61i 167 . 2  |-  (mStat `  T )  =  ran  R
191, 18eqtri 2458 1  |-  S  =  ran  R
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1437    e. wcel 1870   _Vcvv 3087   (/)c0 3767   ran crn 4855   ` cfv 5601  mStRedcmsr 29900  mStatcmsta 29901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-msta 29921
This theorem is referenced by:  msrid  29971  msrfo  29972  mstapst  29973  elmsta  29974
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