Theorem mvhfval 35746

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 6842	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3		mvhfval.v	. . . . . 6 ⊢ 𝑉 = (mVR‘𝑇)
4	2, 3	eqtr4di 2790	. . . . 5 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5		fveq2 6842	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
6		mvhfval.y	. . . . . . . 8 ⊢ 𝑌 = (mType‘𝑇)
7	5, 6	eqtr4di 2790	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
8	7	fveq1d 6844	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((mType‘𝑡)‘𝑣) = (𝑌‘𝑣))
9	8	opeq1d 4837	. . . . 5 ⊢ (𝑡 = 𝑇 → ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩ = ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
10	4, 9	mpteq12dv 5187	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
11		df-mvh 35705	. . . 4 ⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
12	10, 11, 3	mptfvmpt 7184	. . 3 ⊢ (𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
13		mpt0 6642	. . . . 5 ⊢ (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩) = ∅
14	13	eqcomi 2746	. . . 4 ⊢ ∅ = (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
15		fvprc 6834	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = ∅)
16		fvprc 6834	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅)
17	3, 16	eqtrid 2784	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅)
18	17	mpteq1d 5190	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩) = (𝑣 ∈ ∅ ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩))
19	14, 15, 18	3eqtr4a 2798	. . 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 2760	1 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  ⟨cop 4588   ↦ cmpt 5181  ‘cfv 6500  ⟨“cs1 14531  mVRcmvar 35674  mTypecmty 35675  mVHcmvh 35685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-mvh 35705

This theorem is referenced by:  mvhval  35747  mvhf  35771