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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 2818 | . . . 4 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
2 | mtyf2.v | . . . 4 ⊢ 𝑉 = (mVR‘𝑇) | |
3 | mtyf2.y | . . . 4 ⊢ 𝑌 = (mType‘𝑇) | |
4 | eqid 2818 | . . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇) | |
5 | mvtf2.k | . . . 4 ⊢ 𝐾 = (mTC‘𝑇) | |
6 | eqid 2818 | . . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇) | |
7 | eqid 2818 | . . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ismfs 32693 | . . 3 ⊢ (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡𝑌 “ {𝑣}) ∈ Fin)))) |
9 | 8 | ibi 268 | . 2 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡𝑌 “ {𝑣}) ∈ Fin))) |
10 | 9 | simplrd 766 | 1 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 {csn 4557 ◡ccnv 5547 “ cima 5551 ⟶wf 6344 ‘cfv 6348 Fincfn 8497 mCNcmcn 32604 mVRcmvar 32605 mTypecmty 32606 mVTcmvt 32607 mTCcmtc 32608 mAxcmax 32609 mStatcmsta 32619 mFScmfs 32620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-mfs 32640 |
This theorem is referenced by: mtyf 32696 mvtss 32697 msubff1 32700 mvhf 32702 msubvrs 32704 |
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