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Mirrors > Home > MPE Home > Th. List > Mathboxes > mstaval | Structured version Visualization version GIF version |
Description: Value of the set of statements. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mstaval.r | β’ π = (mStRedβπ) |
mstaval.s | β’ π = (mStatβπ) |
Ref | Expression |
---|---|
mstaval | β’ π = ran π |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mstaval.s | . 2 β’ π = (mStatβπ) | |
2 | fveq2 6882 | . . . . . 6 β’ (π‘ = π β (mStRedβπ‘) = (mStRedβπ)) | |
3 | mstaval.r | . . . . . 6 β’ π = (mStRedβπ) | |
4 | 2, 3 | eqtr4di 2782 | . . . . 5 β’ (π‘ = π β (mStRedβπ‘) = π ) |
5 | 4 | rneqd 5928 | . . . 4 β’ (π‘ = π β ran (mStRedβπ‘) = ran π ) |
6 | df-msta 34977 | . . . 4 β’ mStat = (π‘ β V β¦ ran (mStRedβπ‘)) | |
7 | 3 | fvexi 6896 | . . . . 5 β’ π β V |
8 | 7 | rnex 7897 | . . . 4 β’ ran π β V |
9 | 5, 6, 8 | fvmpt 6989 | . . 3 β’ (π β V β (mStatβπ) = ran π ) |
10 | rn0 5916 | . . . . 5 β’ ran β = β | |
11 | 10 | eqcomi 2733 | . . . 4 β’ β = ran β |
12 | fvprc 6874 | . . . 4 β’ (Β¬ π β V β (mStatβπ) = β ) | |
13 | fvprc 6874 | . . . . . 6 β’ (Β¬ π β V β (mStRedβπ) = β ) | |
14 | 3, 13 | eqtrid 2776 | . . . . 5 β’ (Β¬ π β V β π = β ) |
15 | 14 | rneqd 5928 | . . . 4 β’ (Β¬ π β V β ran π = ran β ) |
16 | 11, 12, 15 | 3eqtr4a 2790 | . . 3 β’ (Β¬ π β V β (mStatβπ) = ran π ) |
17 | 9, 16 | pm2.61i 182 | . 2 β’ (mStatβπ) = ran π |
18 | 1, 17 | eqtri 2752 | 1 β’ π = ran π |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1533 β wcel 2098 Vcvv 3466 β c0 4315 ran crn 5668 βcfv 6534 mStRedcmsr 34956 mStatcmsta 34957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-iota 6486 df-fun 6536 df-fv 6542 df-msta 34977 |
This theorem is referenced by: msrid 35027 msrfo 35028 mstapst 35029 elmsta 35030 |
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