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Theorem mvtinf 33026
 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 2759 . . . . 5 (mCN‘𝑇) = (mCN‘𝑇)
2 eqid 2759 . . . . 5 (mVR‘𝑇) = (mVR‘𝑇)
3 mvtinf.y . . . . 5 𝑌 = (mType‘𝑇)
4 mvtinf.f . . . . 5 𝐹 = (mVT‘𝑇)
5 eqid 2759 . . . . 5 (mTC‘𝑇) = (mTC‘𝑇)
6 eqid 2759 . . . . 5 (mAx‘𝑇) = (mAx‘𝑇)
7 eqid 2759 . . . . 5 (mStat‘𝑇) = (mStat‘𝑇)
81, 2, 3, 4, 5, 6, 7ismfs 33020 . . . 4 (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ 𝑌:(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin))))
98ibi 270 . . 3 (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ 𝑌:(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin)))
109simprrd 774 . 2 (𝑇 ∈ mFS → ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin)
11 sneq 4533 . . . . . 6 (𝑣 = 𝑋 → {𝑣} = {𝑋})
1211imaeq2d 5902 . . . . 5 (𝑣 = 𝑋 → (𝑌 “ {𝑣}) = (𝑌 “ {𝑋}))
1312eleq1d 2837 . . . 4 (𝑣 = 𝑋 → ((𝑌 “ {𝑣}) ∈ Fin ↔ (𝑌 “ {𝑋}) ∈ Fin))
1413notbid 322 . . 3 (𝑣 = 𝑋 → (¬ (𝑌 “ {𝑣}) ∈ Fin ↔ ¬ (𝑌 “ {𝑋}) ∈ Fin))
1514rspccva 3541 . 2 ((∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin ∧ 𝑋𝐹) → ¬ (𝑌 “ {𝑋}) ∈ Fin)
1610, 15sylan 584 1 ((𝑇 ∈ mFS ∧ 𝑋𝐹) → ¬ (𝑌 “ {𝑋}) ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071   ∩ cin 3858   ⊆ wss 3859  ∅c0 4226  {csn 4523  ◡ccnv 5524   “ cima 5528  ⟶wf 6332  ‘cfv 6336  Fincfn 8528  mCNcmcn 32931  mVRcmvar 32932  mTypecmty 32933  mVTcmvt 32934  mTCcmtc 32935  mAxcmax 32936  mStatcmsta 32946  mFScmfs 32947 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rab 3080  df-v 3412  df-un 3864  df-in 3866  df-ss 3876  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-mfs 32967 This theorem is referenced by: (None)
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