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Theorem mvhfval 35517
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 6906 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3 mvhfval.v . . . . . 6 𝑉 = (mVR‘𝑇)
42, 3eqtr4di 2792 . . . . 5 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5 fveq2 6906 . . . . . . . 8 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
6 mvhfval.y . . . . . . . 8 𝑌 = (mType‘𝑇)
75, 6eqtr4di 2792 . . . . . . 7 (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
87fveq1d 6908 . . . . . 6 (𝑡 = 𝑇 → ((mType‘𝑡)‘𝑣) = (𝑌𝑣))
98opeq1d 4883 . . . . 5 (𝑡 = 𝑇 → ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩ = ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
104, 9mpteq12dv 5238 . . . 4 (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
11 df-mvh 35476 . . . 4 mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
1210, 11, 3mptfvmpt 7247 . . 3 (𝑇 ∈ V → (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
13 mpt0 6710 . . . . 5 (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩) = ∅
1413eqcomi 2743 . . . 4 ∅ = (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
15 fvprc 6898 . . . 4 𝑇 ∈ V → (mVH‘𝑇) = ∅)
16 fvprc 6898 . . . . . 6 𝑇 ∈ V → (mVR‘𝑇) = ∅)
173, 16eqtrid 2786 . . . . 5 𝑇 ∈ V → 𝑉 = ∅)
1817mpteq1d 5242 . . . 4 𝑇 ∈ V → (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
1914, 15, 183eqtr4a 2800 . . 3 𝑇 ∈ V → (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
2012, 19pm2.61i 182 . 2 (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
211, 20eqtri 2762 1 𝐻 = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1536  wcel 2105  Vcvv 3477  c0 4338  cop 4636  cmpt 5230  cfv 6562  ⟨“cs1 14629  mVRcmvar 35445  mTypecmty 35446  mVHcmvh 35456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-mvh 35476
This theorem is referenced by:  mvhval  35518  mvhf  35542
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