Theorem mtyf2 36043

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:𝑉⟶𝐾)

Proof of Theorem mtyf2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . 4 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
2		mtyf2.v	. . . 4 ⊢ 𝑉 = (mVR‘𝑇)
3		mtyf2.y	. . . 4 ⊢ 𝑌 = (mType‘𝑇)
4		eqid 2763	. . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇)
5		mvtf2.k	. . . 4 ⊢ 𝐾 = (mTC‘𝑇)
6		eqid 2763	. . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
7		eqid 2763	. . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
8	1, 2, 3, 4, 5, 6, 7	ismfs 36041	. . 3 ⊢ (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡𝑌 “ {𝑣}) ∈ Fin))))
9	8	ibi 270	. 2 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡𝑌 “ {𝑣}) ∈ Fin)))
10	9	simplrd 781	1 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐾)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  {csn 4589  ^◡ccnv 5660   “ cima 5664  ⟶wf 6532  ‘cfv 6536  Fincfn 8939  mCNcmcn 35952  mVRcmvar 35953  mTypecmty 35954  mVTcmvt 35955  mTCcmtc 35956  mAxcmax 35957  mStatcmsta 35967  mFScmfs 35968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-mfs 35988

This theorem is referenced by:  mtyf  36044  mvtss  36045  msubff1  36048  mvhf  36050  msubvrs  36052