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Theorem mtyf 35537
Description: The type function maps variables to variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mtyf.v 𝑉 = (mVR‘𝑇)
mtyf.f 𝐹 = (mVT‘𝑇)
mtyf.y 𝑌 = (mType‘𝑇)
Assertion
Ref Expression
mtyf (𝑇 ∈ mFS → 𝑌:𝑉𝐹)

Proof of Theorem mtyf
StepHypRef Expression
1 mtyf.v . . . 4 𝑉 = (mVR‘𝑇)
2 eqid 2735 . . . 4 (mTC‘𝑇) = (mTC‘𝑇)
3 mtyf.y . . . 4 𝑌 = (mType‘𝑇)
41, 2, 3mtyf2 35536 . . 3 (𝑇 ∈ mFS → 𝑌:𝑉⟶(mTC‘𝑇))
5 ffn 6737 . . . 4 (𝑌:𝑉⟶(mTC‘𝑇) → 𝑌 Fn 𝑉)
6 dffn4 6827 . . . 4 (𝑌 Fn 𝑉𝑌:𝑉onto→ran 𝑌)
75, 6sylib 218 . . 3 (𝑌:𝑉⟶(mTC‘𝑇) → 𝑌:𝑉onto→ran 𝑌)
8 fof 6821 . . 3 (𝑌:𝑉onto→ran 𝑌𝑌:𝑉⟶ran 𝑌)
94, 7, 83syl 18 . 2 (𝑇 ∈ mFS → 𝑌:𝑉⟶ran 𝑌)
10 mtyf.f . . . 4 𝐹 = (mVT‘𝑇)
1110, 3mvtval 35485 . . 3 𝐹 = ran 𝑌
12 feq3 6719 . . 3 (𝐹 = ran 𝑌 → (𝑌:𝑉𝐹𝑌:𝑉⟶ran 𝑌))
1311, 12ax-mp 5 . 2 (𝑌:𝑉𝐹𝑌:𝑉⟶ran 𝑌)
149, 13sylibr 234 1 (𝑇 ∈ mFS → 𝑌:𝑉𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  ran crn 5690   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  mVRcmvar 35446  mTypecmty 35447  mVTcmvt 35448  mTCcmtc 35449  mFScmfs 35461
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-mvt 35470  df-mfs 35481
This theorem is referenced by: (None)
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