Theorem mvhfval 35527

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 6861	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3		mvhfval.v	. . . . . 6 ⊢ 𝑉 = (mVR‘𝑇)
4	2, 3	eqtr4di 2783	. . . . 5 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5		fveq2 6861	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
6		mvhfval.y	. . . . . . . 8 ⊢ 𝑌 = (mType‘𝑇)
7	5, 6	eqtr4di 2783	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
8	7	fveq1d 6863	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((mType‘𝑡)‘𝑣) = (𝑌‘𝑣))
9	8	opeq1d 4846	. . . . 5 ⊢ (𝑡 = 𝑇 → ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩ = ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
10	4, 9	mpteq12dv 5197	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
11		df-mvh 35486	. . . 4 ⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
12	10, 11, 3	mptfvmpt 7205	. . 3 ⊢ (𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
13		mpt0 6663	. . . . 5 ⊢ (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩) = ∅
14	13	eqcomi 2739	. . . 4 ⊢ ∅ = (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
15		fvprc 6853	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = ∅)
16		fvprc 6853	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅)
17	3, 16	eqtrid 2777	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅)
18	17	mpteq1d 5200	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
19	14, 15, 18	3eqtr4a 2791	. . 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 2753	1 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ∅c0 4299  ⟨cop 4598   ↦ cmpt 5191  ‘cfv 6514  ⟨“cs1 14567  mVRcmvar 35455  mTypecmty 35456  mVHcmvh 35466

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-mvh 35486

This theorem is referenced by:  mvhval  35528  mvhf  35552