Theorem mvtinf 35753

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 2737	. . . . 5 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
2		eqid 2737	. . . . 5 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
3		mvtinf.y	. . . . 5 ⊢ 𝑌 = (mType‘𝑇)
4		mvtinf.f	. . . . 5 ⊢ 𝐹 = (mVT‘𝑇)
5		eqid 2737	. . . . 5 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
6		eqid 2737	. . . . 5 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
7		eqid 2737	. . . . 5 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
8	1, 2, 3, 4, 5, 6, 7	ismfs 35747	. . . 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 267	. . 3 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ 𝑌:(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin)))
10	9	simprrd 774	. 2 ⊢ (𝑇 ∈ mFS → ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin)
11		sneq 4578	. . . . . 6 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋})
12	11	imaeq2d 6019	. . . . 5 ⊢ (𝑣 = 𝑋 → (◡𝑌 “ {𝑣}) = (◡𝑌 “ {𝑋}))
13	12	eleq1d 2822	. . . 4 ⊢ (𝑣 = 𝑋 → ((◡𝑌 “ {𝑣}) ∈ Fin ↔ (◡𝑌 “ {𝑋}) ∈ Fin))
14	13	notbid 318	. . 3 ⊢ (𝑣 = 𝑋 → (¬ (◡𝑌 “ {𝑣}) ∈ Fin ↔ ¬ (◡𝑌 “ {𝑋}) ∈ Fin))
15	14	rspccva 3564	. 2 ⊢ ((∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin)
16	10, 15	sylan 581	1 ⊢ ((𝑇 ∈ mFS ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  {csn 4568  ^◡ccnv 5623   “ cima 5627  ⟶wf 6488  ‘cfv 6492  Fincfn 8886  mCNcmcn 35658  mVRcmvar 35659  mTypecmty 35660  mVTcmvt 35661  mTCcmtc 35662  mAxcmax 35663  mStatcmsta 35673  mFScmfs 35674

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-mfs 35694

This theorem is referenced by: (None)