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Theorem mvtss 32788
 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 2819 . . 3 (mType‘𝑇) = (mType‘𝑇)
31, 2mvtval 32735 . 2 𝐹 = ran (mType‘𝑇)
4 eqid 2819 . . . 4 (mVR‘𝑇) = (mVR‘𝑇)
5 mvtss.k . . . 4 𝐾 = (mTC‘𝑇)
64, 5, 2mtyf2 32786 . . 3 (𝑇 ∈ mFS → (mType‘𝑇):(mVR‘𝑇)⟶𝐾)
76frnd 6514 . 2 (𝑇 ∈ mFS → ran (mType‘𝑇) ⊆ 𝐾)
83, 7eqsstrid 4013 1 (𝑇 ∈ mFS → 𝐹𝐾)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1530   ∈ wcel 2107   ⊆ wss 3934  ran crn 5549  ‘cfv 6348  mVRcmvar 32696  mTypecmty 32697  mVTcmvt 32698  mTCcmtc 32699  mFScmfs 32711 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-mvt 32720  df-mfs 32731 This theorem is referenced by: (None)
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