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

Theorem mvhfval 33495
Description: Value of the function mapping variables to their corresponding variable expressions. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mvhfval.v 𝑉 = (mVR‘𝑇)
mvhfval.y 𝑌 = (mType‘𝑇)
mvhfval.h 𝐻 = (mVH‘𝑇)
Assertion
Ref Expression
mvhfval 𝐻 = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
Distinct variable groups:   𝑣,𝑇   𝑣,𝑉   𝑣,𝑌
Allowed substitution hint:   𝐻(𝑣)

Proof of Theorem mvhfval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 mvhfval.h . 2 𝐻 = (mVH‘𝑇)
2 fveq2 6774 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3 mvhfval.v . . . . . 6 𝑉 = (mVR‘𝑇)
42, 3eqtr4di 2796 . . . . 5 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5 fveq2 6774 . . . . . . . 8 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
6 mvhfval.y . . . . . . . 8 𝑌 = (mType‘𝑇)
75, 6eqtr4di 2796 . . . . . . 7 (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
87fveq1d 6776 . . . . . 6 (𝑡 = 𝑇 → ((mType‘𝑡)‘𝑣) = (𝑌𝑣))
98opeq1d 4810 . . . . 5 (𝑡 = 𝑇 → ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩ = ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
104, 9mpteq12dv 5165 . . . 4 (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
11 df-mvh 33454 . . . 4 mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
1210, 11, 3mptfvmpt 7104 . . 3 (𝑇 ∈ V → (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
13 mpt0 6575 . . . . 5 (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩) = ∅
1413eqcomi 2747 . . . 4 ∅ = (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
15 fvprc 6766 . . . 4 𝑇 ∈ V → (mVH‘𝑇) = ∅)
16 fvprc 6766 . . . . . 6 𝑇 ∈ V → (mVR‘𝑇) = ∅)
173, 16eqtrid 2790 . . . . 5 𝑇 ∈ V → 𝑉 = ∅)
1817mpteq1d 5169 . . . 4 𝑇 ∈ V → (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
1914, 15, 183eqtr4a 2804 . . 3 𝑇 ∈ V → (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
2012, 19pm2.61i 182 . 2 (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
211, 20eqtri 2766 1 𝐻 = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1539  wcel 2106  Vcvv 3432  c0 4256  cop 4567  cmpt 5157  cfv 6433  ⟨“cs1 14300  mVRcmvar 33423  mTypecmty 33424  mVHcmvh 33434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-mvh 33454
This theorem is referenced by:  mvhval  33496  mvhf  33520
  Copyright terms: Public domain W3C validator