Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mvtinf Structured version   Visualization version   GIF version

Theorem mvtinf 31157
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 ∧ 𝑋𝐹) → ¬ (𝑌 “ {𝑋}) ∈ Fin)

Proof of Theorem mvtinf
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . . 5 (mCN‘𝑇) = (mCN‘𝑇)
2 eqid 2621 . . . . 5 (mVR‘𝑇) = (mVR‘𝑇)
3 mvtinf.y . . . . 5 𝑌 = (mType‘𝑇)
4 mvtinf.f . . . . 5 𝐹 = (mVT‘𝑇)
5 eqid 2621 . . . . 5 (mTC‘𝑇) = (mTC‘𝑇)
6 eqid 2621 . . . . 5 (mAx‘𝑇) = (mAx‘𝑇)
7 eqid 2621 . . . . 5 (mStat‘𝑇) = (mStat‘𝑇)
81, 2, 3, 4, 5, 6, 7ismfs 31151 . . . 4 (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ 𝑌:(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin))))
98ibi 256 . . 3 (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ 𝑌:(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin)))
109simprrd 796 . 2 (𝑇 ∈ mFS → ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin)
11 sneq 4158 . . . . . 6 (𝑣 = 𝑋 → {𝑣} = {𝑋})
1211imaeq2d 5425 . . . . 5 (𝑣 = 𝑋 → (𝑌 “ {𝑣}) = (𝑌 “ {𝑋}))
1312eleq1d 2683 . . . 4 (𝑣 = 𝑋 → ((𝑌 “ {𝑣}) ∈ Fin ↔ (𝑌 “ {𝑋}) ∈ Fin))
1413notbid 308 . . 3 (𝑣 = 𝑋 → (¬ (𝑌 “ {𝑣}) ∈ Fin ↔ ¬ (𝑌 “ {𝑋}) ∈ Fin))
1514rspccva 3294 . 2 ((∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin ∧ 𝑋𝐹) → ¬ (𝑌 “ {𝑋}) ∈ Fin)
1610, 15sylan 488 1 ((𝑇 ∈ mFS ∧ 𝑋𝐹) → ¬ (𝑌 “ {𝑋}) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  cin 3554  wss 3555  c0 3891  {csn 4148  ccnv 5073  cima 5077  wf 5843  cfv 5847  Fincfn 7899  mCNcmcn 31062  mVRcmvar 31063  mTypecmty 31064  mVTcmvt 31065  mTCcmtc 31066  mAxcmax 31067  mStatcmsta 31077  mFScmfs 31078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-mfs 31098
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator