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 = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“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.

Step	Hyp	Ref	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)
4	2, 3	eqtr4di 2796	. . . . 5 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5		fveq2 6774	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
6		mvhfval.y	. . . . . . . 8 ⊢ 𝑌 = (mType‘𝑇)
7	5, 6	eqtr4di 2796	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
8	7	fveq1d 6776	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((mType‘𝑡)‘𝑣) = (𝑌‘𝑣))
9	8	opeq1d 4810	. . . . 5 ⊢ (𝑡 = 𝑇 → ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩ = ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
10	4, 9	mpteq12dv 5165	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
11		df-mvh 33454	. . . 4 ⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
12	10, 11, 3	mptfvmpt 7104	. . 3 ⊢ (𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
13		mpt0 6575	. . . . 5 ⊢ (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩) = ∅
14	13	eqcomi 2747	. . . 4 ⊢ ∅ = (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
15		fvprc 6766	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = ∅)
16		fvprc 6766	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅)
17	3, 16	eqtrid 2790	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅)
18	17	mpteq1d 5169	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
19	14, 15, 18	3eqtr4a 2804	. . 3 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
20	12, 19	pm2.61i 182	. 2 ⊢ (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
21	1, 20	eqtri 2766	1 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)

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