Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mtyf Structured version   Visualization version   GIF version

Theorem mtyf 31756
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 2760 . . . 4 (mTC‘𝑇) = (mTC‘𝑇)
3 mtyf.y . . . 4 𝑌 = (mType‘𝑇)
41, 2, 3mtyf2 31755 . . 3 (𝑇 ∈ mFS → 𝑌:𝑉⟶(mTC‘𝑇))
5 ffn 6206 . . . 4 (𝑌:𝑉⟶(mTC‘𝑇) → 𝑌 Fn 𝑉)
6 dffn4 6282 . . . 4 (𝑌 Fn 𝑉𝑌:𝑉onto→ran 𝑌)
75, 6sylib 208 . . 3 (𝑌:𝑉⟶(mTC‘𝑇) → 𝑌:𝑉onto→ran 𝑌)
8 fof 6276 . . 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 31704 . . 3 𝐹 = ran 𝑌
12 feq3 6189 . . 3 (𝐹 = ran 𝑌 → (𝑌:𝑉𝐹𝑌:𝑉⟶ran 𝑌))
1311, 12ax-mp 5 . 2 (𝑌:𝑉𝐹𝑌:𝑉⟶ran 𝑌)
149, 13sylibr 224 1 (𝑇 ∈ mFS → 𝑌:𝑉𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  ran crn 5267   Fn wfn 6044  wf 6045  ontowfo 6047  cfv 6049  mVRcmvar 31665  mTypecmty 31666  mVTcmvt 31667  mTCcmtc 31668  mFScmfs 31680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-mvt 31689  df-mfs 31700
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator