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Mirrors > Home > MPE Home > Th. List > Mathboxes > mvtinf | Structured version Visualization version GIF version |
Description: Each variable typecode has infinitely many variables. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mvtinf.f | ⊢ 𝐹 = (mVT‘𝑇) |
mvtinf.y | ⊢ 𝑌 = (mType‘𝑇) |
Ref | Expression |
---|---|
mvtinf | ⊢ ((𝑇 ∈ mFS ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin) |
Step | Hyp | Ref | 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) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ismfs 34371 | . . . 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 266 | . . 3 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ 𝑌:(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin))) |
10 | 9 | simprrd 772 | . 2 ⊢ (𝑇 ∈ mFS → ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin) |
11 | sneq 4632 | . . . . . 6 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
12 | 11 | imaeq2d 6049 | . . . . 5 ⊢ (𝑣 = 𝑋 → (◡𝑌 “ {𝑣}) = (◡𝑌 “ {𝑋})) |
13 | 12 | eleq1d 2817 | . . . 4 ⊢ (𝑣 = 𝑋 → ((◡𝑌 “ {𝑣}) ∈ Fin ↔ (◡𝑌 “ {𝑋}) ∈ Fin)) |
14 | 13 | notbid 317 | . . 3 ⊢ (𝑣 = 𝑋 → (¬ (◡𝑌 “ {𝑣}) ∈ Fin ↔ ¬ (◡𝑌 “ {𝑋}) ∈ Fin)) |
15 | 14 | rspccva 3608 | . 2 ⊢ ((∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin) |
16 | 10, 15 | sylan 580 | 1 ⊢ ((𝑇 ∈ mFS ∧ 𝑋 ∈ 𝐹) → ¬ (◡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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