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Theorem mvtinf 31578
 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 2651 . . . . 5 (mCN‘𝑇) = (mCN‘𝑇)
2 eqid 2651 . . . . 5 (mVR‘𝑇) = (mVR‘𝑇)
3 mvtinf.y . . . . 5 𝑌 = (mType‘𝑇)
4 mvtinf.f . . . . 5 𝐹 = (mVT‘𝑇)
5 eqid 2651 . . . . 5 (mTC‘𝑇) = (mTC‘𝑇)
6 eqid 2651 . . . . 5 (mAx‘𝑇) = (mAx‘𝑇)
7 eqid 2651 . . . . 5 (mStat‘𝑇) = (mStat‘𝑇)
81, 2, 3, 4, 5, 6, 7ismfs 31572 . . . 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 812 . 2 (𝑇 ∈ mFS → ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin)
11 sneq 4220 . . . . . 6 (𝑣 = 𝑋 → {𝑣} = {𝑋})
1211imaeq2d 5501 . . . . 5 (𝑣 = 𝑋 → (𝑌 “ {𝑣}) = (𝑌 “ {𝑋}))
1312eleq1d 2715 . . . 4 (𝑣 = 𝑋 → ((𝑌 “ {𝑣}) ∈ Fin ↔ (𝑌 “ {𝑋}) ∈ Fin))
1413notbid 307 . . 3 (𝑣 = 𝑋 → (¬ (𝑌 “ {𝑣}) ∈ Fin ↔ ¬ (𝑌 “ {𝑋}) ∈ Fin))
1514rspccva 3339 . 2 ((∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin ∧ 𝑋𝐹) → ¬ (𝑌 “ {𝑋}) ∈ Fin)
1610, 15sylan 487 1 ((𝑇 ∈ mFS ∧ 𝑋𝐹) → ¬ (𝑌 “ {𝑋}) ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  {csn 4210  ◡ccnv 5142   “ cima 5146  ⟶wf 5922  ‘cfv 5926  Fincfn 7997  mCNcmcn 31483  mVRcmvar 31484  mTypecmty 31485  mVTcmvt 31486  mTCcmtc 31487  mAxcmax 31488  mStatcmsta 31498  mFScmfs 31499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-mfs 31519 This theorem is referenced by: (None)
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