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Theorem mtyf2 31990
 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 2825 . . . 4 (mCN‘𝑇) = (mCN‘𝑇)
2 mtyf2.v . . . 4 𝑉 = (mVR‘𝑇)
3 mtyf2.y . . . 4 𝑌 = (mType‘𝑇)
4 eqid 2825 . . . 4 (mVT‘𝑇) = (mVT‘𝑇)
5 mvtf2.k . . . 4 𝐾 = (mTC‘𝑇)
6 eqid 2825 . . . 4 (mAx‘𝑇) = (mAx‘𝑇)
7 eqid 2825 . . . 4 (mStat‘𝑇) = (mStat‘𝑇)
81, 2, 3, 4, 5, 6, 7ismfs 31988 . . 3 (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (𝑌 “ {𝑣}) ∈ Fin))))
98ibi 259 . 2 (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (𝑌 “ {𝑣}) ∈ Fin)))
109simplrd 786 1 (𝑇 ∈ mFS → 𝑌:𝑉𝐾)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∀wral 3117   ∩ cin 3797   ⊆ wss 3798  ∅c0 4146  {csn 4399  ◡ccnv 5345   “ cima 5349  ⟶wf 6123  ‘cfv 6127  Fincfn 8228  mCNcmcn 31899  mVRcmvar 31900  mTypecmty 31901  mVTcmvt 31902  mTCcmtc 31903  mAxcmax 31904  mStatcmsta 31914  mFScmfs 31915 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-mfs 31935 This theorem is referenced by:  mtyf  31991  mvtss  31992  msubff1  31995  mvhf  31997  msubvrs  31999
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