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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 6919 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇)) | |
3 | mstaval.r | . . . . . 6 ⊢ 𝑅 = (mStRed‘𝑇) | |
4 | 2, 3 | eqtr4di 2792 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅) |
5 | 4 | rneqd 5962 | . . . 4 ⊢ (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅) |
6 | df-msta 35455 | . . . 4 ⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡)) | |
7 | 3 | fvexi 6933 | . . . . 5 ⊢ 𝑅 ∈ V |
8 | 7 | rnex 7946 | . . . 4 ⊢ ran 𝑅 ∈ V |
9 | 5, 6, 8 | fvmpt 7027 | . . 3 ⊢ (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅) |
10 | rn0 5949 | . . . . 5 ⊢ ran ∅ = ∅ | |
11 | 10 | eqcomi 2743 | . . . 4 ⊢ ∅ = ran ∅ |
12 | fvprc 6911 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ∅) | |
13 | fvprc 6911 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅) | |
14 | 3, 13 | eqtrid 2786 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅) |
15 | 14 | rneqd 5962 | . . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑅 = ran ∅) |
16 | 11, 12, 15 | 3eqtr4a 2800 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅) |
17 | 9, 16 | pm2.61i 182 | . 2 ⊢ (mStat‘𝑇) = ran 𝑅 |
18 | 1, 17 | eqtri 2762 | 1 ⊢ 𝑆 = ran 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2103 Vcvv 3482 ∅c0 4347 ran crn 5700 ‘cfv 6572 mStRedcmsr 35434 mStatcmsta 35435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-iota 6524 df-fun 6574 df-fv 6580 df-msta 35455 |
This theorem is referenced by: msrid 35505 msrfo 35506 mstapst 35507 elmsta 35508 |
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