Theorem mvhfval 35761

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 6827	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3		mvhfval.v	. . . . . 6 ⊢ 𝑉 = (mVR‘𝑇)
4	2, 3	eqtr4di 2792	. . . . 5 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5		fveq2 6827	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
6		mvhfval.y	. . . . . . . 8 ⊢ 𝑌 = (mType‘𝑇)
7	5, 6	eqtr4di 2792	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
8	7	fveq1d 6829	. . . . . 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 5159	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
11		df-mvh 35720	. . . 4 ⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
12	10, 11, 3	mptfvmpt 7172	. . 3 ⊢ (𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
13		mpt0 6627	. . . . 5 ⊢ (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩) = ∅
14	13	eqcomi 2748	. . . 4 ⊢ ∅ = (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
15		fvprc 6819	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = ∅)
16		fvprc 6819	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅)
17	3, 16	eqtrid 2786	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅)
18	17	mpteq1d 5162	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
19	14, 15, 18	3eqtr4a 2800	. . 3 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
20	12, 19	pm2.61i 183	. 2 ⊢ (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
21	1, 20	eqtri 2762	1 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)

Syntax hints:  ¬ wn 3   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ∅c0 4261  ⟨cop 4561   ↦ cmpt 5153  ‘cfv 6485  ⟨“cs1 14549  mVRcmvar 35689  mTypecmty 35690  mVHcmvh 35700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-mvh 35720

This theorem is referenced by:  mvhval  35762  mvhf  35786