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Mathbox for Mario Carneiro |
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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 2798 | . . . 4 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
2 | mtyf2.v | . . . 4 ⊢ 𝑉 = (mVR‘𝑇) | |
3 | mtyf2.y | . . . 4 ⊢ 𝑌 = (mType‘𝑇) | |
4 | eqid 2798 | . . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇) | |
5 | mvtf2.k | . . . 4 ⊢ 𝐾 = (mTC‘𝑇) | |
6 | eqid 2798 | . . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇) | |
7 | eqid 2798 | . . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ismfs 32909 | . . 3 ⊢ (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡𝑌 “ {𝑣}) ∈ Fin)))) |
9 | 8 | ibi 270 | . 2 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡𝑌 “ {𝑣}) ∈ Fin))) |
10 | 9 | simplrd 769 | 1 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 ◡ccnv 5518 “ cima 5522 ⟶wf 6320 ‘cfv 6324 Fincfn 8492 mCNcmcn 32820 mVRcmvar 32821 mTypecmty 32822 mVTcmvt 32823 mTCcmtc 32824 mAxcmax 32825 mStatcmsta 32835 mFScmfs 32836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-mfs 32856 |
This theorem is referenced by: mtyf 32912 mvtss 32913 msubff1 32916 mvhf 32918 msubvrs 32920 |
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