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Mirrors > Home > MPE Home > Th. List > Mathboxes > mvtss | Structured version Visualization version GIF version |
Description: The set of variable typecodes is a subset of all typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mvtss.f | ⊢ 𝐹 = (mVT‘𝑇) |
mvtss.k | ⊢ 𝐾 = (mTC‘𝑇) |
Ref | Expression |
---|---|
mvtss | ⊢ (𝑇 ∈ mFS → 𝐹 ⊆ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvtss.f | . . 3 ⊢ 𝐹 = (mVT‘𝑇) | |
2 | eqid 2798 | . . 3 ⊢ (mType‘𝑇) = (mType‘𝑇) | |
3 | 1, 2 | mvtval 32860 | . 2 ⊢ 𝐹 = ran (mType‘𝑇) |
4 | eqid 2798 | . . . 4 ⊢ (mVR‘𝑇) = (mVR‘𝑇) | |
5 | mvtss.k | . . . 4 ⊢ 𝐾 = (mTC‘𝑇) | |
6 | 4, 5, 2 | mtyf2 32911 | . . 3 ⊢ (𝑇 ∈ mFS → (mType‘𝑇):(mVR‘𝑇)⟶𝐾) |
7 | 6 | frnd 6494 | . 2 ⊢ (𝑇 ∈ mFS → ran (mType‘𝑇) ⊆ 𝐾) |
8 | 3, 7 | eqsstrid 3963 | 1 ⊢ (𝑇 ∈ mFS → 𝐹 ⊆ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ran crn 5520 ‘cfv 6324 mVRcmvar 32821 mTypecmty 32822 mVTcmvt 32823 mTCcmtc 32824 mFScmfs 32836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-mvt 32845 df-mfs 32856 |
This theorem is referenced by: (None) |
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