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Theorem mvtinf 34377
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 2731 . . . . 5 (mCN‘𝑇) = (mCN‘𝑇)
2 eqid 2731 . . . . 5 (mVR‘𝑇) = (mVR‘𝑇)
3 mvtinf.y . . . . 5 𝑌 = (mType‘𝑇)
4 mvtinf.f . . . . 5 𝐹 = (mVT‘𝑇)
5 eqid 2731 . . . . 5 (mTC‘𝑇) = (mTC‘𝑇)
6 eqid 2731 . . . . 5 (mAx‘𝑇) = (mAx‘𝑇)
7 eqid 2731 . . . . 5 (mStat‘𝑇) = (mStat‘𝑇)
81, 2, 3, 4, 5, 6, 7ismfs 34371 . . . 4 (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ 𝑌:(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin))))
98ibi 266 . . 3 (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ 𝑌:(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin)))
109simprrd 772 . 2 (𝑇 ∈ mFS → ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin)
11 sneq 4632 . . . . . 6 (𝑣 = 𝑋 → {𝑣} = {𝑋})
1211imaeq2d 6049 . . . . 5 (𝑣 = 𝑋 → (𝑌 “ {𝑣}) = (𝑌 “ {𝑋}))
1312eleq1d 2817 . . . 4 (𝑣 = 𝑋 → ((𝑌 “ {𝑣}) ∈ Fin ↔ (𝑌 “ {𝑋}) ∈ Fin))
1413notbid 317 . . 3 (𝑣 = 𝑋 → (¬ (𝑌 “ {𝑣}) ∈ Fin ↔ ¬ (𝑌 “ {𝑋}) ∈ Fin))
1514rspccva 3608 . 2 ((∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin ∧ 𝑋𝐹) → ¬ (𝑌 “ {𝑋}) ∈ Fin)
1610, 15sylan 580 1 ((𝑇 ∈ mFS ∧ 𝑋𝐹) → ¬ (𝑌 “ {𝑋}) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3060  cin 3943  wss 3944  c0 4318  {csn 4622  ccnv 5668  cima 5672  wf 6528  cfv 6532  Fincfn 8922  mCNcmcn 34282  mVRcmvar 34283  mTypecmty 34284  mVTcmvt 34285  mTCcmtc 34286  mAxcmax 34287  mStatcmsta 34297  mFScmfs 34298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-mfs 34318
This theorem is referenced by: (None)
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