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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mtyf2 | Structured version Visualization version GIF version | ||
| Description: The type function maps variables to typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.) |
| Ref | Expression |
|---|---|
| mtyf2.v | ⊢ 𝑉 = (mVR‘𝑇) |
| mvtf2.k | ⊢ 𝐾 = (mTC‘𝑇) |
| mtyf2.y | ⊢ 𝑌 = (mType‘𝑇) |
| Ref | Expression |
|---|---|
| mtyf2 | ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐾) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
| 2 | mtyf2.v | . . . 4 ⊢ 𝑉 = (mVR‘𝑇) | |
| 3 | mtyf2.y | . . . 4 ⊢ 𝑌 = (mType‘𝑇) | |
| 4 | eqid 2737 | . . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇) | |
| 5 | mvtf2.k | . . . 4 ⊢ 𝐾 = (mTC‘𝑇) | |
| 6 | eqid 2737 | . . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇) | |
| 7 | eqid 2737 | . . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | ismfs 35554 | . . 3 ⊢ (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡𝑌 “ {𝑣}) ∈ Fin)))) |
| 9 | 8 | ibi 267 | . 2 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡𝑌 “ {𝑣}) ∈ Fin))) |
| 10 | 9 | simplrd 770 | 1 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐾) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 ◡ccnv 5684 “ cima 5688 ⟶wf 6557 ‘cfv 6561 Fincfn 8985 mCNcmcn 35465 mVRcmvar 35466 mTypecmty 35467 mVTcmvt 35468 mTCcmtc 35469 mAxcmax 35470 mStatcmsta 35480 mFScmfs 35481 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-mfs 35501 |
| This theorem is referenced by: mtyf 35557 mvtss 35558 msubff1 35561 mvhf 35563 msubvrs 35565 |
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