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Theorem mvhfval 31404
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 6178 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3 mvhfval.v . . . . . 6 𝑉 = (mVR‘𝑇)
42, 3syl6eqr 2672 . . . . 5 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5 fveq2 6178 . . . . . . . 8 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
6 mvhfval.y . . . . . . . 8 𝑌 = (mType‘𝑇)
75, 6syl6eqr 2672 . . . . . . 7 (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
87fveq1d 6180 . . . . . 6 (𝑡 = 𝑇 → ((mType‘𝑡)‘𝑣) = (𝑌𝑣))
98opeq1d 4399 . . . . 5 (𝑡 = 𝑇 → ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩ = ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
104, 9mpteq12dv 4724 . . . 4 (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
11 df-mvh 31363 . . . 4 mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
12 fvex 6188 . . . . . 6 (mVR‘𝑇) ∈ V
133, 12eqeltri 2695 . . . . 5 𝑉 ∈ V
1413mptex 6471 . . . 4 (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩) ∈ V
1510, 11, 14fvmpt 6269 . . 3 (𝑇 ∈ V → (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
16 mpt0 6008 . . . . 5 (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩) = ∅
1716eqcomi 2629 . . . 4 ∅ = (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
18 fvprc 6172 . . . 4 𝑇 ∈ V → (mVH‘𝑇) = ∅)
19 fvprc 6172 . . . . . 6 𝑇 ∈ V → (mVR‘𝑇) = ∅)
203, 19syl5eq 2666 . . . . 5 𝑇 ∈ V → 𝑉 = ∅)
2120mpteq1d 4729 . . . 4 𝑇 ∈ V → (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
2217, 18, 213eqtr4a 2680 . . 3 𝑇 ∈ V → (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
2315, 22pm2.61i 176 . 2 (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
241, 23eqtri 2642 1 𝐻 = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1481  wcel 1988  Vcvv 3195  c0 3907  cop 4174  cmpt 4720  cfv 5876  ⟨“cs1 13277  mVRcmvar 31332  mTypecmty 31333  mVHcmvh 31343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-mvh 31363
This theorem is referenced by:  mvhval  31405  mvhf  31429
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