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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 2821 | . . . 4 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
3 | mtyf.y | . . . 4 ⊢ 𝑌 = (mType‘𝑇) | |
4 | 1, 2, 3 | mtyf2 32793 | . . 3 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶(mTC‘𝑇)) |
5 | ffn 6509 | . . . 4 ⊢ (𝑌:𝑉⟶(mTC‘𝑇) → 𝑌 Fn 𝑉) | |
6 | dffn4 6591 | . . . 4 ⊢ (𝑌 Fn 𝑉 ↔ 𝑌:𝑉–onto→ran 𝑌) | |
7 | 5, 6 | sylib 220 | . . 3 ⊢ (𝑌:𝑉⟶(mTC‘𝑇) → 𝑌:𝑉–onto→ran 𝑌) |
8 | fof 6585 | . . 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 32742 | . . 3 ⊢ 𝐹 = ran 𝑌 |
12 | feq3 6492 | . . 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 236 | 1 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ran crn 5551 Fn wfn 6345 ⟶wf 6346 –onto→wfo 6348 ‘cfv 6350 mVRcmvar 32703 mTypecmty 32704 mVTcmvt 32705 mTCcmtc 32706 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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
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-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fo 6356 df-fv 6358 df-mvt 32727 df-mfs 32738 |
This theorem is referenced by: (None) |
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