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Theorem mvtss 30704
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 2610 . . 3 (mType‘𝑇) = (mType‘𝑇)
31, 2mvtval 30651 . 2 𝐹 = ran (mType‘𝑇)
4 eqid 2610 . . . 4 (mVR‘𝑇) = (mVR‘𝑇)
5 mvtss.k . . . 4 𝐾 = (mTC‘𝑇)
64, 5, 2mtyf2 30702 . . 3 (𝑇 ∈ mFS → (mType‘𝑇):(mVR‘𝑇)⟶𝐾)
7 frn 5966 . . 3 ((mType‘𝑇):(mVR‘𝑇)⟶𝐾 → ran (mType‘𝑇) ⊆ 𝐾)
86, 7syl 17 . 2 (𝑇 ∈ mFS → ran (mType‘𝑇) ⊆ 𝐾)
93, 8syl5eqss 3612 1 (𝑇 ∈ mFS → 𝐹𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  wss 3540  ran crn 5039  wf 5800  cfv 5804  mVRcmvar 30612  mTypecmty 30613  mVTcmvt 30614  mTCcmtc 30615  mFScmfs 30627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-mvt 30636  df-mfs 30647
This theorem is referenced by: (None)
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