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Theorem mvtss 31959
Description: The set of variable typecodes is a subset of all typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mvtss.f 𝐹 = (mVT‘𝑇)
mvtss.k 𝐾 = (mTC‘𝑇)
Assertion
Ref Expression
mvtss (𝑇 ∈ mFS → 𝐹𝐾)

Proof of Theorem mvtss
StepHypRef Expression
1 mvtss.f . . 3 𝐹 = (mVT‘𝑇)
2 eqid 2797 . . 3 (mType‘𝑇) = (mType‘𝑇)
31, 2mvtval 31906 . 2 𝐹 = ran (mType‘𝑇)
4 eqid 2797 . . . 4 (mVR‘𝑇) = (mVR‘𝑇)
5 mvtss.k . . . 4 𝐾 = (mTC‘𝑇)
64, 5, 2mtyf2 31957 . . 3 (𝑇 ∈ mFS → (mType‘𝑇):(mVR‘𝑇)⟶𝐾)
76frnd 6261 . 2 (𝑇 ∈ mFS → ran (mType‘𝑇) ⊆ 𝐾)
83, 7syl5eqss 3843 1 (𝑇 ∈ mFS → 𝐹𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  wss 3767  ran crn 5311  cfv 6099  mVRcmvar 31867  mTypecmty 31868  mVTcmvt 31869  mTCcmtc 31870  mFScmfs 31882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fv 6107  df-mvt 31891  df-mfs 31902
This theorem is referenced by: (None)
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