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Theorem mvhfval 30477
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 6087 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3 mvhfval.v . . . . . 6 𝑉 = (mVR‘𝑇)
42, 3syl6eqr 2661 . . . . 5 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5 fveq2 6087 . . . . . . . 8 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
6 mvhfval.y . . . . . . . 8 𝑌 = (mType‘𝑇)
75, 6syl6eqr 2661 . . . . . . 7 (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
87fveq1d 6089 . . . . . 6 (𝑡 = 𝑇 → ((mType‘𝑡)‘𝑣) = (𝑌𝑣))
98opeq1d 4340 . . . . 5 (𝑡 = 𝑇 → ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩ = ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
104, 9mpteq12dv 4657 . . . 4 (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
11 df-mvh 30436 . . . 4 mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
12 fvex 6097 . . . . . 6 (mVR‘𝑇) ∈ V
133, 12eqeltri 2683 . . . . 5 𝑉 ∈ V
1413mptex 6367 . . . 4 (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩) ∈ V
1510, 11, 14fvmpt 6175 . . 3 (𝑇 ∈ V → (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
16 mpt0 5919 . . . . 5 (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩) = ∅
1716eqcomi 2618 . . . 4 ∅ = (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
18 fvprc 6081 . . . 4 𝑇 ∈ V → (mVH‘𝑇) = ∅)
19 fvprc 6081 . . . . . 6 𝑇 ∈ V → (mVR‘𝑇) = ∅)
203, 19syl5eq 2655 . . . . 5 𝑇 ∈ V → 𝑉 = ∅)
2120mpteq1d 4660 . . . 4 𝑇 ∈ V → (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ ∅ ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
2217, 18, 213eqtr4a 2669 . . 3 𝑇 ∈ V → (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩))
2315, 22pm2.61i 174 . 2 (mVH‘𝑇) = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
241, 23eqtri 2631 1 𝐻 = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1474  wcel 1976  Vcvv 3172  c0 3873  cop 4130  cmpt 4637  cfv 5789  ⟨“cs1 13097  mVRcmvar 30405  mTypecmty 30406  mVHcmvh 30416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-mvh 30436
This theorem is referenced by:  mvhval  30478  mvhf  30502
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