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Theorem mvtss 31778
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 2760 . . 3 (mType‘𝑇) = (mType‘𝑇)
31, 2mvtval 31725 . 2 𝐹 = ran (mType‘𝑇)
4 eqid 2760 . . . 4 (mVR‘𝑇) = (mVR‘𝑇)
5 mvtss.k . . . 4 𝐾 = (mTC‘𝑇)
64, 5, 2mtyf2 31776 . . 3 (𝑇 ∈ mFS → (mType‘𝑇):(mVR‘𝑇)⟶𝐾)
7 frn 6214 . . 3 ((mType‘𝑇):(mVR‘𝑇)⟶𝐾 → ran (mType‘𝑇) ⊆ 𝐾)
86, 7syl 17 . 2 (𝑇 ∈ mFS → ran (mType‘𝑇) ⊆ 𝐾)
93, 8syl5eqss 3790 1 (𝑇 ∈ mFS → 𝐹𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wss 3715  ran crn 5267  wf 6045  cfv 6049  mVRcmvar 31686  mTypecmty 31687  mVTcmvt 31688  mTCcmtc 31689  mFScmfs 31701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-mvt 31710  df-mfs 31721
This theorem is referenced by: (None)
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