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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 2622 | . . . . 5 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
2 | eqid 2622 | . . . . 5 ⊢ (mVR‘𝑇) = (mVR‘𝑇) | |
3 | mvtinf.y | . . . . 5 ⊢ 𝑌 = (mType‘𝑇) | |
4 | mvtinf.f | . . . . 5 ⊢ 𝐹 = (mVT‘𝑇) | |
5 | eqid 2622 | . . . . 5 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
6 | eqid 2622 | . . . . 5 ⊢ (mAx‘𝑇) = (mAx‘𝑇) | |
7 | eqid 2622 | . . . . 5 ⊢ (mStat‘𝑇) = (mStat‘𝑇) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ismfs 31446 | . . . 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 256 | . . 3 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ 𝑌:(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin))) |
10 | 9 | simprrd 797 | . 2 ⊢ (𝑇 ∈ mFS → ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin) |
11 | sneq 4187 | . . . . . 6 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
12 | 11 | imaeq2d 5466 | . . . . 5 ⊢ (𝑣 = 𝑋 → (◡𝑌 “ {𝑣}) = (◡𝑌 “ {𝑋})) |
13 | 12 | eleq1d 2686 | . . . 4 ⊢ (𝑣 = 𝑋 → ((◡𝑌 “ {𝑣}) ∈ Fin ↔ (◡𝑌 “ {𝑋}) ∈ Fin)) |
14 | 13 | notbid 308 | . . 3 ⊢ (𝑣 = 𝑋 → (¬ (◡𝑌 “ {𝑣}) ∈ Fin ↔ ¬ (◡𝑌 “ {𝑋}) ∈ Fin)) |
15 | 14 | rspccva 3308 | . 2 ⊢ ((∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin) |
16 | 10, 15 | sylan 488 | 1 ⊢ ((𝑇 ∈ mFS ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {csn 4177 ◡ccnv 5113 “ cima 5117 ⟶wf 5884 ‘cfv 5888 Fincfn 7955 mCNcmcn 31357 mVRcmvar 31358 mTypecmty 31359 mVTcmvt 31360 mTCcmtc 31361 mAxcmax 31362 mStatcmsta 31372 mFScmfs 31373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-mfs 31393 |
This theorem is referenced by: (None) |
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