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Theorem mtyf2 30702
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 2610 . . . . 5 (mCN‘𝑇) = (mCN‘𝑇)
2 mtyf2.v . . . . 5 𝑉 = (mVR‘𝑇)
3 mtyf2.y . . . . 5 𝑌 = (mType‘𝑇)
4 eqid 2610 . . . . 5 (mVT‘𝑇) = (mVT‘𝑇)
5 mvtf2.k . . . . 5 𝐾 = (mTC‘𝑇)
6 eqid 2610 . . . . 5 (mAx‘𝑇) = (mAx‘𝑇)
7 eqid 2610 . . . . 5 (mStat‘𝑇) = (mStat‘𝑇)
81, 2, 3, 4, 5, 6, 7ismfs 30700 . . . 4 (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (𝑌 “ {𝑣}) ∈ Fin))))
98ibi 255 . . 3 (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (𝑌 “ {𝑣}) ∈ Fin)))
109simpld 474 . 2 (𝑇 ∈ mFS → (((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉𝐾))
1110simprd 478 1 (𝑇 ∈ mFS → 𝑌:𝑉𝐾)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  cin 3539  wss 3540  c0 3874  {csn 4125  ccnv 5037  cima 5041  wf 5800  cfv 5804  Fincfn 7841  mCNcmcn 30611  mVRcmvar 30612  mTypecmty 30613  mVTcmvt 30614  mTCcmtc 30615  mAxcmax 30616  mStatcmsta 30626  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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  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-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-mfs 30647
This theorem is referenced by:  mtyf  30703  mvtss  30704  msubff1  30707  mvhf  30709  msubvrs  30711
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