Theorem mtyf2 36082

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 2765	. . . 4 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
2		mtyf2.v	. . . 4 ⊢ 𝑉 = (mVR‘𝑇)
3		mtyf2.y	. . . 4 ⊢ 𝑌 = (mType‘𝑇)
4		eqid 2765	. . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇)
5		mvtf2.k	. . . 4 ⊢ 𝐾 = (mTC‘𝑇)
6		eqid 2765	. . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
7		eqid 2765	. . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
8	1, 2, 3, 4, 5, 6, 7	ismfs 36080	. . 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 782	1 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐾)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  {csn 4591  ^◡ccnv 5662   “ cima 5666  ⟶wf 6536  ‘cfv 6540  Fincfn 8949  mCNcmcn 35991  mVRcmvar 35992  mTypecmty 35993  mVTcmvt 35994  mTCcmtc 35995  mAxcmax 35996  mStatcmsta 36006  mFScmfs 36007

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-mfs 36027

This theorem is used by:  mtyf  36083  mvtss  36084  msubff1  36087  mvhf  36089  msubvrs  36091