Theorem mvtinf 36060

Description: Each variable typecode has infinitely many variables. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mvtinf.f	⊢ 𝐹 = (mVT‘𝑇)
mvtinf.y	⊢ 𝑌 = (mType‘𝑇)

Assertion

Ref	Expression
mvtinf	⊢ ((𝑇 ∈ mFS ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin)

Proof of Theorem mvtinf

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . 5 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
2		eqid 2765	. . . . 5 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
3		mvtinf.y	. . . . 5 ⊢ 𝑌 = (mType‘𝑇)
4		mvtinf.f	. . . . 5 ⊢ 𝐹 = (mVT‘𝑇)
5		eqid 2765	. . . . 5 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
6		eqid 2765	. . . . 5 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
7		eqid 2765	. . . . 5 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
8	1, 2, 3, 4, 5, 6, 7	ismfs 36054	. . . 4 ⊢ (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ 𝑌:(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin))))
9	8	ibi 270	. . 3 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ 𝑌:(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin)))
10	9	simprrd 786	. 2 ⊢ (𝑇 ∈ mFS → ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin)
11		sneq 4601	. . . . . 6 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋})
12	11	imaeq2d 6064	. . . . 5 ⊢ (𝑣 = 𝑋 → (◡𝑌 “ {𝑣}) = (◡𝑌 “ {𝑋}))
13	12	eleq1d 2850	. . . 4 ⊢ (𝑣 = 𝑋 → ((◡𝑌 “ {𝑣}) ∈ Fin ↔ (◡𝑌 “ {𝑋}) ∈ Fin))
14	13	notbid 321	. . 3 ⊢ (𝑣 = 𝑋 → (¬ (◡𝑌 “ {𝑣}) ∈ Fin ↔ ¬ (◡𝑌 “ {𝑋}) ∈ Fin))
15	14	rspccva 3582	. 2 ⊢ ((∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin)
16	10, 15	sylan 592	1 ⊢ ((𝑇 ∈ mFS ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin)

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 8945  mCNcmcn 35965  mVRcmvar 35966  mTypecmty 35967  mVTcmvt 35968  mTCcmtc 35969  mAxcmax 35970  mStatcmsta 35980  mFScmfs 35981

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-xp 5669  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 36001

This theorem is used by: (None)