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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 6875 . . . . . 6 (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3 mstaval.r . . . . . 6 𝑅 = (mStRed‘𝑇)
42, 3eqtr4di 2788 . . . . 5 (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
54rneqd 5918 . . . 4 (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅)
6 df-msta 35463 . . . 4 mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
73fvexi 6889 . . . . 5 𝑅 ∈ V
87rnex 7904 . . . 4 ran 𝑅 ∈ V
95, 6, 8fvmpt 6985 . . 3 (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
10 rn0 5905 . . . . 5 ran ∅ = ∅
1110eqcomi 2744 . . . 4 ∅ = ran ∅
12 fvprc 6867 . . . 4 𝑇 ∈ V → (mStat‘𝑇) = ∅)
13 fvprc 6867 . . . . . 6 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
143, 13eqtrid 2782 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
1514rneqd 5918 . . . 4 𝑇 ∈ V → ran 𝑅 = ran ∅)
1611, 12, 153eqtr4a 2796 . . 3 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
179, 16pm2.61i 182 . 2 (mStat‘𝑇) = ran 𝑅
181, 17eqtri 2758 1 𝑆 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3459  c0 4308  ran crn 5655  cfv 6530  mStRedcmsr 35442  mStatcmsta 35443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7727
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6483  df-fun 6532  df-fv 6538  df-msta 35463
This theorem is referenced by:  msrid  35513  msrfo  35514  mstapst  35515  elmsta  35516
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