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Theorem mstaval 35512
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 6920 . . . . . 6 (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3 mstaval.r . . . . . 6 𝑅 = (mStRed‘𝑇)
42, 3eqtr4di 2798 . . . . 5 (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
54rneqd 5963 . . . 4 (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅)
6 df-msta 35463 . . . 4 mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
73fvexi 6934 . . . . 5 𝑅 ∈ V
87rnex 7950 . . . 4 ran 𝑅 ∈ V
95, 6, 8fvmpt 7029 . . 3 (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
10 rn0 5950 . . . . 5 ran ∅ = ∅
1110eqcomi 2749 . . . 4 ∅ = ran ∅
12 fvprc 6912 . . . 4 𝑇 ∈ V → (mStat‘𝑇) = ∅)
13 fvprc 6912 . . . . . 6 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
143, 13eqtrid 2792 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
1514rneqd 5963 . . . 4 𝑇 ∈ V → ran 𝑅 = ran ∅)
1611, 12, 153eqtr4a 2806 . . 3 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
179, 16pm2.61i 182 . 2 (mStat‘𝑇) = ran 𝑅
181, 17eqtri 2768 1 𝑆 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2108  Vcvv 3488  c0 4352  ran crn 5701  cfv 6573  mStRedcmsr 35442  mStatcmsta 35443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-msta 35463
This theorem is referenced by:  msrid  35513  msrfo  35514  mstapst  35515  elmsta  35516
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