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Theorem mvhfval 34524
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 6892 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3 mvhfval.v . . . . . 6 𝑉 = (mVR‘𝑇)
42, 3eqtr4di 2791 . . . . 5 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5 fveq2 6892 . . . . . . . 8 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
6 mvhfval.y . . . . . . . 8 𝑌 = (mType‘𝑇)
75, 6eqtr4di 2791 . . . . . . 7 (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
87fveq1d 6894 . . . . . 6 (𝑡 = 𝑇 → ((mType‘𝑡)‘𝑣) = (𝑌𝑣))
98opeq1d 4880 . . . . 5 (𝑡 = 𝑇 → ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩ = ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
104, 9mpteq12dv 5240 . . . 4 (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
11 df-mvh 34483 . . . 4 mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
1210, 11, 3mptfvmpt 7230 . . 3 (𝑇 ∈ V → (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
13 mpt0 6693 . . . . 5 (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩) = ∅
1413eqcomi 2742 . . . 4 ∅ = (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
15 fvprc 6884 . . . 4 𝑇 ∈ V → (mVH‘𝑇) = ∅)
16 fvprc 6884 . . . . . 6 𝑇 ∈ V → (mVR‘𝑇) = ∅)
173, 16eqtrid 2785 . . . . 5 𝑇 ∈ V → 𝑉 = ∅)
1817mpteq1d 5244 . . . 4 𝑇 ∈ V → (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
1914, 15, 183eqtr4a 2799 . . 3 𝑇 ∈ V → (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
2012, 19pm2.61i 182 . 2 (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
211, 20eqtri 2761 1 𝐻 = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3475  c0 4323  cop 4635  cmpt 5232  cfv 6544  ⟨“cs1 14545  mVRcmvar 34452  mTypecmty 34453  mVHcmvh 34463
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-mvh 34483
This theorem is referenced by:  mvhval  34525  mvhf  34549
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