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Theorem mstaval 35504
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 6919 . . . . . 6 (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3 mstaval.r . . . . . 6 𝑅 = (mStRed‘𝑇)
42, 3eqtr4di 2792 . . . . 5 (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
54rneqd 5962 . . . 4 (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅)
6 df-msta 35455 . . . 4 mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
73fvexi 6933 . . . . 5 𝑅 ∈ V
87rnex 7946 . . . 4 ran 𝑅 ∈ V
95, 6, 8fvmpt 7027 . . 3 (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
10 rn0 5949 . . . . 5 ran ∅ = ∅
1110eqcomi 2743 . . . 4 ∅ = ran ∅
12 fvprc 6911 . . . 4 𝑇 ∈ V → (mStat‘𝑇) = ∅)
13 fvprc 6911 . . . . . 6 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
143, 13eqtrid 2786 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
1514rneqd 5962 . . . 4 𝑇 ∈ V → ran 𝑅 = ran ∅)
1611, 12, 153eqtr4a 2800 . . 3 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
179, 16pm2.61i 182 . 2 (mStat‘𝑇) = ran 𝑅
181, 17eqtri 2762 1 𝑆 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2103  Vcvv 3482  c0 4347  ran crn 5700  cfv 6572  mStRedcmsr 35434  mStatcmsta 35435
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-iota 6524  df-fun 6574  df-fv 6580  df-msta 35455
This theorem is referenced by:  msrid  35505  msrfo  35506  mstapst  35507  elmsta  35508
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