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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 6103 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇)) | |
3 | mstaval.r | . . . . . 6 ⊢ 𝑅 = (mStRed‘𝑇) | |
4 | 2, 3 | syl6eqr 2662 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅) |
5 | 4 | rneqd 5274 | . . . 4 ⊢ (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅) |
6 | df-msta 30646 | . . . 4 ⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡)) | |
7 | fvex 6113 | . . . . . 6 ⊢ (mStRed‘𝑇) ∈ V | |
8 | 3, 7 | eqeltri 2684 | . . . . 5 ⊢ 𝑅 ∈ V |
9 | 8 | rnex 6992 | . . . 4 ⊢ ran 𝑅 ∈ V |
10 | 5, 6, 9 | fvmpt 6191 | . . 3 ⊢ (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅) |
11 | rn0 5298 | . . . . 5 ⊢ ran ∅ = ∅ | |
12 | 11 | eqcomi 2619 | . . . 4 ⊢ ∅ = ran ∅ |
13 | fvprc 6097 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ∅) | |
14 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅) | |
15 | 3, 14 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅) |
16 | 15 | rneqd 5274 | . . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑅 = ran ∅) |
17 | 12, 13, 16 | 3eqtr4a 2670 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅) |
18 | 10, 17 | pm2.61i 175 | . 2 ⊢ (mStat‘𝑇) = ran 𝑅 |
19 | 1, 18 | eqtri 2632 | 1 ⊢ 𝑆 = ran 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ran crn 5039 ‘cfv 5804 mStRedcmsr 30625 mStatcmsta 30626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-msta 30646 |
This theorem is referenced by: msrid 30696 msrfo 30697 mstapst 30698 elmsta 30699 |
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