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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 2610 | . . . . 5 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
2 | eqid 2610 | . . . . 5 ⊢ (mVR‘𝑇) = (mVR‘𝑇) | |
3 | mvtinf.y | . . . . 5 ⊢ 𝑌 = (mType‘𝑇) | |
4 | mvtinf.f | . . . . 5 ⊢ 𝐹 = (mVT‘𝑇) | |
5 | eqid 2610 | . . . . 5 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
6 | eqid 2610 | . . . . 5 ⊢ (mAx‘𝑇) = (mAx‘𝑇) | |
7 | eqid 2610 | . . . . 5 ⊢ (mStat‘𝑇) = (mStat‘𝑇) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ismfs 30700 | . . . 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 255 | . . 3 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ 𝑌:(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin))) |
10 | 9 | simprrd 793 | . 2 ⊢ (𝑇 ∈ mFS → ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin) |
11 | sneq 4135 | . . . . . 6 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
12 | 11 | imaeq2d 5385 | . . . . 5 ⊢ (𝑣 = 𝑋 → (◡𝑌 “ {𝑣}) = (◡𝑌 “ {𝑋})) |
13 | 12 | eleq1d 2672 | . . . 4 ⊢ (𝑣 = 𝑋 → ((◡𝑌 “ {𝑣}) ∈ Fin ↔ (◡𝑌 “ {𝑋}) ∈ Fin)) |
14 | 13 | notbid 307 | . . 3 ⊢ (𝑣 = 𝑋 → (¬ (◡𝑌 “ {𝑣}) ∈ Fin ↔ ¬ (◡𝑌 “ {𝑋}) ∈ Fin)) |
15 | 14 | rspccva 3281 | . 2 ⊢ ((∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin) |
16 | 10, 15 | sylan 487 | 1 ⊢ ((𝑇 ∈ mFS ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 ◡ccnv 5037 “ cima 5041 ⟶wf 5800 ‘cfv 5804 Fincfn 7841 mCNcmcn 30611 mVRcmvar 30612 mTypecmty 30613 mVTcmvt 30614 mTCcmtc 30615 mAxcmax 30616 mStatcmsta 30626 mFScmfs 30627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-mfs 30647 |
This theorem is referenced by: (None) |
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