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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 6672 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇)) | |
3 | mstaval.r | . . . . . 6 ⊢ 𝑅 = (mStRed‘𝑇) | |
4 | 2, 3 | syl6eqr 2876 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅) |
5 | 4 | rneqd 5810 | . . . 4 ⊢ (𝑡 = 𝑇 → ran (mStRed‘𝑡) = ran 𝑅) |
6 | df-msta 32744 | . . . 4 ⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡)) | |
7 | 3 | fvexi 6686 | . . . . 5 ⊢ 𝑅 ∈ V |
8 | 7 | rnex 7619 | . . . 4 ⊢ ran 𝑅 ∈ V |
9 | 5, 6, 8 | fvmpt 6770 | . . 3 ⊢ (𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅) |
10 | rn0 5798 | . . . . 5 ⊢ ran ∅ = ∅ | |
11 | 10 | eqcomi 2832 | . . . 4 ⊢ ∅ = ran ∅ |
12 | fvprc 6665 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ∅) | |
13 | fvprc 6665 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅) | |
14 | 3, 13 | syl5eq 2870 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅) |
15 | 14 | rneqd 5810 | . . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑅 = ran ∅) |
16 | 11, 12, 15 | 3eqtr4a 2884 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mStat‘𝑇) = ran 𝑅) |
17 | 9, 16 | pm2.61i 184 | . 2 ⊢ (mStat‘𝑇) = ran 𝑅 |
18 | 1, 17 | eqtri 2846 | 1 ⊢ 𝑆 = ran 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 ran crn 5558 ‘cfv 6357 mStRedcmsr 32723 mStatcmsta 32724 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fv 6365 df-msta 32744 |
This theorem is referenced by: msrid 32794 msrfo 32795 mstapst 32796 elmsta 32797 |
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