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Theorem mstaval 31146
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 6148 . . . . . 6 (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3 mstaval.r . . . . . 6 𝑅 = (mStRed‘𝑇)
42, 3syl6eqr 2673 . . . . 5 (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
54rneqd 5313 . . . 4 (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅)
6 df-msta 31097 . . . 4 mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
7 fvex 6158 . . . . . 6 (mStRed‘𝑇) ∈ V
83, 7eqeltri 2694 . . . . 5 𝑅 ∈ V
98rnex 7047 . . . 4 ran 𝑅 ∈ V
105, 6, 9fvmpt 6239 . . 3 (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
11 rn0 5337 . . . . 5 ran ∅ = ∅
1211eqcomi 2630 . . . 4 ∅ = ran ∅
13 fvprc 6142 . . . 4 𝑇 ∈ V → (mStat‘𝑇) = ∅)
14 fvprc 6142 . . . . . 6 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
153, 14syl5eq 2667 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
1615rneqd 5313 . . . 4 𝑇 ∈ V → ran 𝑅 = ran ∅)
1712, 13, 163eqtr4a 2681 . . 3 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
1810, 17pm2.61i 176 . 2 (mStat‘𝑇) = ran 𝑅
191, 18eqtri 2643 1 𝑆 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3186  c0 3891  ran crn 5075  cfv 5847  mStRedcmsr 31076  mStatcmsta 31077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fv 5855  df-msta 31097
This theorem is referenced by:  msrid  31147  msrfo  31148  mstapst  31149  elmsta  31150
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