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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 | ⊢ 𝑅 = (mStRed‘𝑇) |
mstaval.s | ⊢ 𝑆 = (mStat‘𝑇) |
Ref | Expression |
---|---|
mstaval | ⊢ 𝑆 = ran 𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mstaval.s | . 2 ⊢ 𝑆 = (mStat‘𝑇) | |
2 | fveq2 6891 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇)) | |
3 | mstaval.r | . . . . . 6 ⊢ 𝑅 = (mStRed‘𝑇) | |
4 | 2, 3 | eqtr4di 2789 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅) |
5 | 4 | rneqd 5937 | . . . 4 ⊢ (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅) |
6 | df-msta 34949 | . . . 4 ⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡)) | |
7 | 3 | fvexi 6905 | . . . . 5 ⊢ 𝑅 ∈ V |
8 | 7 | rnex 7907 | . . . 4 ⊢ ran 𝑅 ∈ V |
9 | 5, 6, 8 | fvmpt 6998 | . . 3 ⊢ (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅) |
10 | rn0 5925 | . . . . 5 ⊢ ran ∅ = ∅ | |
11 | 10 | eqcomi 2740 | . . . 4 ⊢ ∅ = ran ∅ |
12 | fvprc 6883 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ∅) | |
13 | fvprc 6883 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅) | |
14 | 3, 13 | eqtrid 2783 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅) |
15 | 14 | rneqd 5937 | . . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑅 = ran ∅) |
16 | 11, 12, 15 | 3eqtr4a 2797 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅) |
17 | 9, 16 | pm2.61i 182 | . 2 ⊢ (mStat‘𝑇) = ran 𝑅 |
18 | 1, 17 | eqtri 2759 | 1 ⊢ 𝑆 = ran 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ∅c0 4322 ran crn 5677 ‘cfv 6543 mStRedcmsr 34928 mStatcmsta 34929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fv 6551 df-msta 34949 |
This theorem is referenced by: msrid 34999 msrfo 35000 mstapst 35001 elmsta 35002 |
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