Theorem mtyf2 33413

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 2738	. . . 4 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
2		mtyf2.v	. . . 4 ⊢ 𝑉 = (mVR‘𝑇)
3		mtyf2.y	. . . 4 ⊢ 𝑌 = (mType‘𝑇)
4		eqid 2738	. . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇)
5		mvtf2.k	. . . 4 ⊢ 𝐾 = (mTC‘𝑇)
6		eqid 2738	. . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
7		eqid 2738	. . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
8	1, 2, 3, 4, 5, 6, 7	ismfs 33411	. . 3 ⊢ (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡𝑌 “ {𝑣}) ∈ Fin))))
9	8	ibi 266	. 2 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡𝑌 “ {𝑣}) ∈ Fin)))
10	9	simplrd 766	1 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐾)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558  ^◡ccnv 5579   “ cima 5583  ⟶wf 6414  ‘cfv 6418  Fincfn 8691  mCNcmcn 33322  mVRcmvar 33323  mTypecmty 33324  mVTcmvt 33325  mTCcmtc 33326  mAxcmax 33327  mStatcmsta 33337  mFScmfs 33338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-mfs 33358

This theorem is referenced by:  mtyf  33414  mvtss  33415  msubff1  33418  mvhf  33420  msubvrs  33422