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Mirrors > Home > MPE Home > Th. List > Mathboxes > mtyf | Structured version Visualization version GIF version |
Description: The type function maps variables to variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mtyf.v | ⊢ 𝑉 = (mVR‘𝑇) |
mtyf.f | ⊢ 𝐹 = (mVT‘𝑇) |
mtyf.y | ⊢ 𝑌 = (mType‘𝑇) |
Ref | Expression |
---|---|
mtyf | ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mtyf.v | . . . 4 ⊢ 𝑉 = (mVR‘𝑇) | |
2 | eqid 2798 | . . . 4 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
3 | mtyf.y | . . . 4 ⊢ 𝑌 = (mType‘𝑇) | |
4 | 1, 2, 3 | mtyf2 32911 | . . 3 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶(mTC‘𝑇)) |
5 | ffn 6487 | . . . 4 ⊢ (𝑌:𝑉⟶(mTC‘𝑇) → 𝑌 Fn 𝑉) | |
6 | dffn4 6571 | . . . 4 ⊢ (𝑌 Fn 𝑉 ↔ 𝑌:𝑉–onto→ran 𝑌) | |
7 | 5, 6 | sylib 221 | . . 3 ⊢ (𝑌:𝑉⟶(mTC‘𝑇) → 𝑌:𝑉–onto→ran 𝑌) |
8 | fof 6565 | . . 3 ⊢ (𝑌:𝑉–onto→ran 𝑌 → 𝑌:𝑉⟶ran 𝑌) | |
9 | 4, 7, 8 | 3syl 18 | . 2 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶ran 𝑌) |
10 | mtyf.f | . . . 4 ⊢ 𝐹 = (mVT‘𝑇) | |
11 | 10, 3 | mvtval 32860 | . . 3 ⊢ 𝐹 = ran 𝑌 |
12 | feq3 6470 | . . 3 ⊢ (𝐹 = ran 𝑌 → (𝑌:𝑉⟶𝐹 ↔ 𝑌:𝑉⟶ran 𝑌)) | |
13 | 11, 12 | ax-mp 5 | . 2 ⊢ (𝑌:𝑉⟶𝐹 ↔ 𝑌:𝑉⟶ran 𝑌) |
14 | 9, 13 | sylibr 237 | 1 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ran crn 5520 Fn wfn 6319 ⟶wf 6320 –onto→wfo 6322 ‘cfv 6324 mVRcmvar 32821 mTypecmty 32822 mVTcmvt 32823 mTCcmtc 32824 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 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
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-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 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-fo 6330 df-fv 6332 df-mvt 32845 df-mfs 32856 |
This theorem is referenced by: (None) |
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