Theorem mvhval 36065

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
mvhval	⊢ (𝑋 ∈ 𝑉 → (𝐻‘𝑋) = ⟨(𝑌‘𝑋), ⟨“𝑋”⟩⟩)

Proof of Theorem mvhval

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . 3 ⊢ (𝑣 = 𝑋 → (𝑌‘𝑣) = (𝑌‘𝑋))
2		s1eq 14659	. . 3 ⊢ (𝑣 = 𝑋 → ⟨“𝑣”⟩ = ⟨“𝑋”⟩)
3	1, 2	opeq12d 4848	. 2 ⊢ (𝑣 = 𝑋 → ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩ = ⟨(𝑌‘𝑋), ⟨“𝑋”⟩⟩)
4		mvhfval.v	. . 3 ⊢ 𝑉 = (mVR‘𝑇)
5		mvhfval.y	. . 3 ⊢ 𝑌 = (mType‘𝑇)
6		mvhfval.h	. . 3 ⊢ 𝐻 = (mVH‘𝑇)
7	4, 5, 6	mvhfval 36064	. 2 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
8		opex 5447	. 2 ⊢ ⟨(𝑌‘𝑋), ⟨“𝑋”⟩⟩ ∈ V
9	3, 7, 8	fvmpt 6993	1 ⊢ (𝑋 ∈ 𝑉 → (𝐻‘𝑋) = ⟨(𝑌‘𝑋), ⟨“𝑋”⟩⟩)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ⟨cop 4597  ‘cfv 6540  ⟨“cs1 14654  mVRcmvar 35992  mTypecmty 35993  mVHcmvh 36003

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-s1 14655  df-mvh 36023

This theorem is used by:  mvhf1  36090  msubvrs  36091