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Theorem mstaval 31770
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 6354 . . . . . 6 (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3 mstaval.r . . . . . 6 𝑅 = (mStRed‘𝑇)
42, 3syl6eqr 2813 . . . . 5 (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
54rneqd 5509 . . . 4 (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅)
6 df-msta 31721 . . . 4 mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
7 fvex 6364 . . . . . 6 (mStRed‘𝑇) ∈ V
83, 7eqeltri 2836 . . . . 5 𝑅 ∈ V
98rnex 7267 . . . 4 ran 𝑅 ∈ V
105, 6, 9fvmpt 6446 . . 3 (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
11 rn0 5533 . . . . 5 ran ∅ = ∅
1211eqcomi 2770 . . . 4 ∅ = ran ∅
13 fvprc 6348 . . . 4 𝑇 ∈ V → (mStat‘𝑇) = ∅)
14 fvprc 6348 . . . . . 6 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
153, 14syl5eq 2807 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
1615rneqd 5509 . . . 4 𝑇 ∈ V → ran 𝑅 = ran ∅)
1712, 13, 163eqtr4a 2821 . . 3 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅)
1810, 17pm2.61i 176 . 2 (mStat‘𝑇) = ran 𝑅
191, 18eqtri 2783 1 𝑆 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1632  wcel 2140  Vcvv 3341  c0 4059  ran crn 5268  cfv 6050  mStRedcmsr 31700  mStatcmsta 31701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-iota 6013  df-fun 6052  df-fv 6058  df-msta 31721
This theorem is referenced by:  msrid  31771  msrfo  31772  mstapst  31773  elmsta  31774
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