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Theorem mstaval 34202
Description: Value of the set of statements. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mstaval.r 𝑅 = (mStRedβ€˜π‘‡)
mstaval.s 𝑆 = (mStatβ€˜π‘‡)
Assertion
Ref Expression
mstaval 𝑆 = ran 𝑅

Proof of Theorem mstaval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 mstaval.s . 2 𝑆 = (mStatβ€˜π‘‡)
2 fveq2 6846 . . . . . 6 (𝑑 = 𝑇 β†’ (mStRedβ€˜π‘‘) = (mStRedβ€˜π‘‡))
3 mstaval.r . . . . . 6 𝑅 = (mStRedβ€˜π‘‡)
42, 3eqtr4di 2791 . . . . 5 (𝑑 = 𝑇 β†’ (mStRedβ€˜π‘‘) = 𝑅)
54rneqd 5897 . . . 4 (𝑑 = 𝑇 β†’ ran (mStRedβ€˜π‘‘) = ran 𝑅)
6 df-msta 34153 . . . 4 mStat = (𝑑 ∈ V ↦ ran (mStRedβ€˜π‘‘))
73fvexi 6860 . . . . 5 𝑅 ∈ V
87rnex 7853 . . . 4 ran 𝑅 ∈ V
95, 6, 8fvmpt 6952 . . 3 (𝑇 ∈ V β†’ (mStatβ€˜π‘‡) = ran 𝑅)
10 rn0 5885 . . . . 5 ran βˆ… = βˆ…
1110eqcomi 2742 . . . 4 βˆ… = ran βˆ…
12 fvprc 6838 . . . 4 (Β¬ 𝑇 ∈ V β†’ (mStatβ€˜π‘‡) = βˆ…)
13 fvprc 6838 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mStRedβ€˜π‘‡) = βˆ…)
143, 13eqtrid 2785 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝑅 = βˆ…)
1514rneqd 5897 . . . 4 (Β¬ 𝑇 ∈ V β†’ ran 𝑅 = ran βˆ…)
1611, 12, 153eqtr4a 2799 . . 3 (Β¬ 𝑇 ∈ V β†’ (mStatβ€˜π‘‡) = ran 𝑅)
179, 16pm2.61i 182 . 2 (mStatβ€˜π‘‡) = ran 𝑅
181, 17eqtri 2761 1 𝑆 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1542   ∈ wcel 2107  Vcvv 3447  βˆ…c0 4286  ran crn 5638  β€˜cfv 6500  mStRedcmsr 34132  mStatcmsta 34133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fv 6508  df-msta 34153
This theorem is referenced by:  msrid  34203  msrfo  34204  mstapst  34205  elmsta  34206
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