Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mstaval Structured version   Visualization version   GIF version

Theorem mstaval 35149
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 6892 . . . . . 6 (𝑑 = 𝑇 β†’ (mStRedβ€˜π‘‘) = (mStRedβ€˜π‘‡))
3 mstaval.r . . . . . 6 𝑅 = (mStRedβ€˜π‘‡)
42, 3eqtr4di 2786 . . . . 5 (𝑑 = 𝑇 β†’ (mStRedβ€˜π‘‘) = 𝑅)
54rneqd 5935 . . . 4 (𝑑 = 𝑇 β†’ ran (mStRedβ€˜π‘‘) = ran 𝑅)
6 df-msta 35100 . . . 4 mStat = (𝑑 ∈ V ↦ ran (mStRedβ€˜π‘‘))
73fvexi 6906 . . . . 5 𝑅 ∈ V
87rnex 7913 . . . 4 ran 𝑅 ∈ V
95, 6, 8fvmpt 7000 . . 3 (𝑇 ∈ V β†’ (mStatβ€˜π‘‡) = ran 𝑅)
10 rn0 5923 . . . . 5 ran βˆ… = βˆ…
1110eqcomi 2737 . . . 4 βˆ… = ran βˆ…
12 fvprc 6884 . . . 4 (Β¬ 𝑇 ∈ V β†’ (mStatβ€˜π‘‡) = βˆ…)
13 fvprc 6884 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mStRedβ€˜π‘‡) = βˆ…)
143, 13eqtrid 2780 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝑅 = βˆ…)
1514rneqd 5935 . . . 4 (Β¬ 𝑇 ∈ V β†’ ran 𝑅 = ran βˆ…)
1611, 12, 153eqtr4a 2794 . . 3 (Β¬ 𝑇 ∈ V β†’ (mStatβ€˜π‘‡) = ran 𝑅)
179, 16pm2.61i 182 . 2 (mStatβ€˜π‘‡) = ran 𝑅
181, 17eqtri 2756 1 𝑆 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1534   ∈ wcel 2099  Vcvv 3470  βˆ…c0 4319  ran crn 5674  β€˜cfv 6543  mStRedcmsr 35079  mStatcmsta 35080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6495  df-fun 6545  df-fv 6551  df-msta 35100
This theorem is referenced by:  msrid  35150  msrfo  35151  mstapst  35152  elmsta  35153
  Copyright terms: Public domain W3C validator