Theorem mvtinf 35918

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 35912	. . . 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 785	. 2 ⊢ (𝑇 ∈ mFS → ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin)
11		sneq 4595	. . . . . 6 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋})
12	11	imaeq2d 6053	. . . . 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 3583	. 2 ⊢ ((∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin)
16	10, 15	sylan 591	1 ⊢ ((𝑇 ∈ mFS ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  {csn 4585  ^◡ccnv 5651   “ cima 5655  ⟶wf 6521  ‘cfv 6525  Fincfn 8931  mCNcmcn 35823  mVRcmvar 35824  mTypecmty 35825  mVTcmvt 35826  mTCcmtc 35827  mAxcmax 35828  mStatcmsta 35838  mFScmfs 35839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-mfs 35859

This theorem is referenced by: (None)