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Theorem mtyf 32807
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 2820 . . . 4 (mTC‘𝑇) = (mTC‘𝑇)
3 mtyf.y . . . 4 𝑌 = (mType‘𝑇)
41, 2, 3mtyf2 32806 . . 3 (𝑇 ∈ mFS → 𝑌:𝑉⟶(mTC‘𝑇))
5 ffn 6490 . . . 4 (𝑌:𝑉⟶(mTC‘𝑇) → 𝑌 Fn 𝑉)
6 dffn4 6572 . . . 4 (𝑌 Fn 𝑉𝑌:𝑉onto→ran 𝑌)
75, 6sylib 220 . . 3 (𝑌:𝑉⟶(mTC‘𝑇) → 𝑌:𝑉onto→ran 𝑌)
8 fof 6566 . . 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 32755 . . 3 𝐹 = ran 𝑌
12 feq3 6473 . . 3 (𝐹 = ran 𝑌 → (𝑌:𝑉𝐹𝑌:𝑉⟶ran 𝑌))
1311, 12ax-mp 5 . 2 (𝑌:𝑉𝐹𝑌:𝑉⟶ran 𝑌)
149, 13sylibr 236 1 (𝑇 ∈ mFS → 𝑌:𝑉𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1537  wcel 2114  ran crn 5532   Fn wfn 6326  wf 6327  ontowfo 6329  cfv 6331  mVRcmvar 32716  mTypecmty 32717  mVTcmvt 32718  mTCcmtc 32719  mFScmfs 32731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-fo 6337  df-fv 6339  df-mvt 32740  df-mfs 32751
This theorem is referenced by: (None)
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