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Theorem mstaval 34530
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 6891 . . . . . 6 (𝑑 = 𝑇 β†’ (mStRedβ€˜π‘‘) = (mStRedβ€˜π‘‡))
3 mstaval.r . . . . . 6 𝑅 = (mStRedβ€˜π‘‡)
42, 3eqtr4di 2790 . . . . 5 (𝑑 = 𝑇 β†’ (mStRedβ€˜π‘‘) = 𝑅)
54rneqd 5937 . . . 4 (𝑑 = 𝑇 β†’ ran (mStRedβ€˜π‘‘) = ran 𝑅)
6 df-msta 34481 . . . 4 mStat = (𝑑 ∈ V ↦ ran (mStRedβ€˜π‘‘))
73fvexi 6905 . . . . 5 𝑅 ∈ V
87rnex 7902 . . . 4 ran 𝑅 ∈ V
95, 6, 8fvmpt 6998 . . 3 (𝑇 ∈ V β†’ (mStatβ€˜π‘‡) = ran 𝑅)
10 rn0 5925 . . . . 5 ran βˆ… = βˆ…
1110eqcomi 2741 . . . 4 βˆ… = ran βˆ…
12 fvprc 6883 . . . 4 (Β¬ 𝑇 ∈ V β†’ (mStatβ€˜π‘‡) = βˆ…)
13 fvprc 6883 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mStRedβ€˜π‘‡) = βˆ…)
143, 13eqtrid 2784 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝑅 = βˆ…)
1514rneqd 5937 . . . 4 (Β¬ 𝑇 ∈ V β†’ ran 𝑅 = ran βˆ…)
1611, 12, 153eqtr4a 2798 . . 3 (Β¬ 𝑇 ∈ V β†’ (mStatβ€˜π‘‡) = ran 𝑅)
179, 16pm2.61i 182 . 2 (mStatβ€˜π‘‡) = ran 𝑅
181, 17eqtri 2760 1 𝑆 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1541   ∈ wcel 2106  Vcvv 3474  βˆ…c0 4322  ran crn 5677  β€˜cfv 6543  mStRedcmsr 34460  mStatcmsta 34461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-msta 34481
This theorem is referenced by:  msrid  34531  msrfo  34532  mstapst  34533  elmsta  34534
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