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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 2821 | . . . 4 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
2 | mtyf2.v | . . . 4 ⊢ 𝑉 = (mVR‘𝑇) | |
3 | mtyf2.y | . . . 4 ⊢ 𝑌 = (mType‘𝑇) | |
4 | eqid 2821 | . . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇) | |
5 | mvtf2.k | . . . 4 ⊢ 𝐾 = (mTC‘𝑇) | |
6 | eqid 2821 | . . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇) | |
7 | eqid 2821 | . . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ismfs 32791 | . . 3 ⊢ (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡𝑌 “ {𝑣}) ∈ Fin)))) |
9 | 8 | ibi 269 | . 2 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡𝑌 “ {𝑣}) ∈ Fin))) |
10 | 9 | simplrd 768 | 1 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 {csn 4560 ◡ccnv 5548 “ cima 5552 ⟶wf 6345 ‘cfv 6349 Fincfn 8503 mCNcmcn 32702 mVRcmvar 32703 mTypecmty 32704 mVTcmvt 32705 mTCcmtc 32706 mAxcmax 32707 mStatcmsta 32717 mFScmfs 32718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-mfs 32738 |
This theorem is referenced by: mtyf 32794 mvtss 32795 msubff1 32798 mvhf 32800 msubvrs 32802 |
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