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Theorem mtyf2 35536
Description: The type function maps variables to typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mtyf2.v 𝑉 = (mVR‘𝑇)
mvtf2.k 𝐾 = (mTC‘𝑇)
mtyf2.y 𝑌 = (mType‘𝑇)
Assertion
Ref Expression
mtyf2 (𝑇 ∈ mFS → 𝑌:𝑉𝐾)

Proof of Theorem mtyf2
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . . 4 (mCN‘𝑇) = (mCN‘𝑇)
2 mtyf2.v . . . 4 𝑉 = (mVR‘𝑇)
3 mtyf2.y . . . 4 𝑌 = (mType‘𝑇)
4 eqid 2735 . . . 4 (mVT‘𝑇) = (mVT‘𝑇)
5 mvtf2.k . . . 4 𝐾 = (mTC‘𝑇)
6 eqid 2735 . . . 4 (mAx‘𝑇) = (mAx‘𝑇)
7 eqid 2735 . . . 4 (mStat‘𝑇) = (mStat‘𝑇)
81, 2, 3, 4, 5, 6, 7ismfs 35534 . . 3 (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (𝑌 “ {𝑣}) ∈ Fin))))
98ibi 267 . 2 (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (𝑌 “ {𝑣}) ∈ Fin)))
109simplrd 770 1 (𝑇 ∈ mFS → 𝑌:𝑉𝐾)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  cin 3962  wss 3963  c0 4339  {csn 4631  ccnv 5688  cima 5692  wf 6559  cfv 6563  Fincfn 8984  mCNcmcn 35445  mVRcmvar 35446  mTypecmty 35447  mVTcmvt 35448  mTCcmtc 35449  mAxcmax 35450  mStatcmsta 35460  mFScmfs 35461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-mfs 35481
This theorem is referenced by:  mtyf  35537  mvtss  35538  msubff1  35541  mvhf  35543  msubvrs  35545
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