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Theorem mstaval 34998
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 6891 . . . . . 6 (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3 mstaval.r . . . . . 6 𝑅 = (mStRed‘𝑇)
42, 3eqtr4di 2789 . . . . 5 (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
54rneqd 5937 . . . 4 (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅)
6 df-msta 34949 . . . 4 mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
73fvexi 6905 . . . . 5 𝑅 ∈ V
87rnex 7907 . . . 4 ran 𝑅 ∈ V
95, 6, 8fvmpt 6998 . . 3 (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
10 rn0 5925 . . . . 5 ran ∅ = ∅
1110eqcomi 2740 . . . 4 ∅ = ran ∅
12 fvprc 6883 . . . 4 𝑇 ∈ V → (mStat‘𝑇) = ∅)
13 fvprc 6883 . . . . . 6 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
143, 13eqtrid 2783 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
1514rneqd 5937 . . . 4 𝑇 ∈ V → ran 𝑅 = ran ∅)
1611, 12, 153eqtr4a 2797 . . 3 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
179, 16pm2.61i 182 . 2 (mStat‘𝑇) = ran 𝑅
181, 17eqtri 2759 1 𝑆 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2105  Vcvv 3473  c0 4322  ran crn 5677  cfv 6543  mStRedcmsr 34928  mStatcmsta 34929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-msta 34949
This theorem is referenced by:  msrid  34999  msrfo  35000  mstapst  35001  elmsta  35002
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