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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 6846 | . . . . . 6 β’ (π‘ = π β (mStRedβπ‘) = (mStRedβπ)) | |
3 | mstaval.r | . . . . . 6 β’ π = (mStRedβπ) | |
4 | 2, 3 | eqtr4di 2791 | . . . . 5 β’ (π‘ = π β (mStRedβπ‘) = π ) |
5 | 4 | rneqd 5897 | . . . 4 β’ (π‘ = π β ran (mStRedβπ‘) = ran π ) |
6 | df-msta 34153 | . . . 4 β’ mStat = (π‘ β V β¦ ran (mStRedβπ‘)) | |
7 | 3 | fvexi 6860 | . . . . 5 β’ π β V |
8 | 7 | rnex 7853 | . . . 4 β’ ran π β V |
9 | 5, 6, 8 | fvmpt 6952 | . . 3 β’ (π β V β (mStatβπ) = ran π ) |
10 | rn0 5885 | . . . . 5 β’ ran β = β | |
11 | 10 | eqcomi 2742 | . . . 4 β’ β = ran β |
12 | fvprc 6838 | . . . 4 β’ (Β¬ π β V β (mStatβπ) = β ) | |
13 | fvprc 6838 | . . . . . 6 β’ (Β¬ π β V β (mStRedβπ) = β ) | |
14 | 3, 13 | eqtrid 2785 | . . . . 5 β’ (Β¬ π β V β π = β ) |
15 | 14 | rneqd 5897 | . . . 4 β’ (Β¬ π β V β ran π = ran β ) |
16 | 11, 12, 15 | 3eqtr4a 2799 | . . 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 2761 | 1 β’ π = ran π |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1542 β wcel 2107 Vcvv 3447 β c0 4286 ran crn 5638 βcfv 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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