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Theorem mstaval 30695
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 6103 . . . . . 6 (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3 mstaval.r . . . . . 6 𝑅 = (mStRed‘𝑇)
42, 3syl6eqr 2662 . . . . 5 (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
54rneqd 5274 . . . 4 (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅)
6 df-msta 30646 . . . 4 mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
7 fvex 6113 . . . . . 6 (mStRed‘𝑇) ∈ V
83, 7eqeltri 2684 . . . . 5 𝑅 ∈ V
98rnex 6992 . . . 4 ran 𝑅 ∈ V
105, 6, 9fvmpt 6191 . . 3 (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
11 rn0 5298 . . . . 5 ran ∅ = ∅
1211eqcomi 2619 . . . 4 ∅ = ran ∅
13 fvprc 6097 . . . 4 𝑇 ∈ V → (mStat‘𝑇) = ∅)
14 fvprc 6097 . . . . . 6 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
153, 14syl5eq 2656 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
1615rneqd 5274 . . . 4 𝑇 ∈ V → ran 𝑅 = ran ∅)
1712, 13, 163eqtr4a 2670 . . 3 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
1810, 17pm2.61i 175 . 2 (mStat‘𝑇) = ran 𝑅
191, 18eqtri 2632 1 𝑆 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1475  wcel 1977  Vcvv 3173  c0 3874  ran crn 5039  cfv 5804  mStRedcmsr 30625  mStatcmsta 30626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-msta 30646
This theorem is referenced by:  msrid  30696  msrfo  30697  mstapst  30698  elmsta  30699
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