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Theorem mstaval 35026
Description: Value of the set of statements. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mstaval.r 𝑅 = (mStRedβ€˜π‘‡)
mstaval.s 𝑆 = (mStatβ€˜π‘‡)
Assertion
Ref Expression
mstaval 𝑆 = ran 𝑅

Proof of Theorem mstaval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 mstaval.s . 2 𝑆 = (mStatβ€˜π‘‡)
2 fveq2 6882 . . . . . 6 (𝑑 = 𝑇 β†’ (mStRedβ€˜π‘‘) = (mStRedβ€˜π‘‡))
3 mstaval.r . . . . . 6 𝑅 = (mStRedβ€˜π‘‡)
42, 3eqtr4di 2782 . . . . 5 (𝑑 = 𝑇 β†’ (mStRedβ€˜π‘‘) = 𝑅)
54rneqd 5928 . . . 4 (𝑑 = 𝑇 β†’ ran (mStRedβ€˜π‘‘) = ran 𝑅)
6 df-msta 34977 . . . 4 mStat = (𝑑 ∈ V ↦ ran (mStRedβ€˜π‘‘))
73fvexi 6896 . . . . 5 𝑅 ∈ V
87rnex 7897 . . . 4 ran 𝑅 ∈ V
95, 6, 8fvmpt 6989 . . 3 (𝑇 ∈ V β†’ (mStatβ€˜π‘‡) = ran 𝑅)
10 rn0 5916 . . . . 5 ran βˆ… = βˆ…
1110eqcomi 2733 . . . 4 βˆ… = ran βˆ…
12 fvprc 6874 . . . 4 (Β¬ 𝑇 ∈ V β†’ (mStatβ€˜π‘‡) = βˆ…)
13 fvprc 6874 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mStRedβ€˜π‘‡) = βˆ…)
143, 13eqtrid 2776 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝑅 = βˆ…)
1514rneqd 5928 . . . 4 (Β¬ 𝑇 ∈ V β†’ ran 𝑅 = ran βˆ…)
1611, 12, 153eqtr4a 2790 . . 3 (Β¬ 𝑇 ∈ V β†’ (mStatβ€˜π‘‡) = ran 𝑅)
179, 16pm2.61i 182 . 2 (mStatβ€˜π‘‡) = ran 𝑅
181, 17eqtri 2752 1 𝑆 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1533   ∈ wcel 2098  Vcvv 3466  βˆ…c0 4315  ran crn 5668  β€˜cfv 6534  mStRedcmsr 34956  mStatcmsta 34957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-msta 34977
This theorem is referenced by:  msrid  35027  msrfo  35028  mstapst  35029  elmsta  35030
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