Theorem mtyf2 35556

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 2737	. . . 4 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
2		mtyf2.v	. . . 4 ⊢ 𝑉 = (mVR‘𝑇)
3		mtyf2.y	. . . 4 ⊢ 𝑌 = (mType‘𝑇)
4		eqid 2737	. . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇)
5		mvtf2.k	. . . 4 ⊢ 𝐾 = (mTC‘𝑇)
6		eqid 2737	. . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
7		eqid 2737	. . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
8	1, 2, 3, 4, 5, 6, 7	ismfs 35554	. . 3 ⊢ (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡𝑌 “ {𝑣}) ∈ Fin))))
9	8	ibi 267	. 2 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡𝑌 “ {𝑣}) ∈ Fin)))
10	9	simplrd 770	1 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐾)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  {csn 4626  ^◡ccnv 5684   “ cima 5688  ⟶wf 6557  ‘cfv 6561  Fincfn 8985  mCNcmcn 35465  mVRcmvar 35466  mTypecmty 35467  mVTcmvt 35468  mTCcmtc 35469  mAxcmax 35470  mStatcmsta 35480  mFScmfs 35481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mfs 35501

This theorem is referenced by:  mtyf  35557  mvtss  35558  msubff1  35561  mvhf  35563  msubvrs  35565