Theorem mvhfval 35538

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 6906	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3		mvhfval.v	. . . . . 6 ⊢ 𝑉 = (mVR‘𝑇)
4	2, 3	eqtr4di 2795	. . . . 5 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5		fveq2 6906	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
6		mvhfval.y	. . . . . . . 8 ⊢ 𝑌 = (mType‘𝑇)
7	5, 6	eqtr4di 2795	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
8	7	fveq1d 6908	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((mType‘𝑡)‘𝑣) = (𝑌‘𝑣))
9	8	opeq1d 4879	. . . . 5 ⊢ (𝑡 = 𝑇 → ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩ = ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
10	4, 9	mpteq12dv 5233	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
11		df-mvh 35497	. . . 4 ⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
12	10, 11, 3	mptfvmpt 7248	. . 3 ⊢ (𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
13		mpt0 6710	. . . . 5 ⊢ (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩) = ∅
14	13	eqcomi 2746	. . . 4 ⊢ ∅ = (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
15		fvprc 6898	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = ∅)
16		fvprc 6898	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅)
17	3, 16	eqtrid 2789	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅)
18	17	mpteq1d 5237	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
19	14, 15, 18	3eqtr4a 2803	. . 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 2765	1 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ∅c0 4333  ⟨cop 4632   ↦ cmpt 5225  ‘cfv 6561  ⟨“cs1 14633  mVRcmvar 35466  mTypecmty 35467  mVHcmvh 35477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-mvh 35497

This theorem is referenced by:  mvhval  35539  mvhf  35563