Theorem mvhval 35730

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 6834	. . 3 ⊢ (𝑣 = 𝑋 → (𝑌‘𝑣) = (𝑌‘𝑋))
2		s1eq 14526	. . 3 ⊢ (𝑣 = 𝑋 → ⟨“𝑣”⟩ = ⟨“𝑋”⟩)
3	1, 2	opeq12d 4837	. 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 35729	. 2 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
8		opex 5412	. 2 ⊢ ⟨(𝑌‘𝑋), ⟨“𝑋”⟩⟩ ∈ V
9	3, 7, 8	fvmpt 6941	1 ⊢ (𝑋 ∈ 𝑉 → (𝐻‘𝑋) = ⟨(𝑌‘𝑋), ⟨“𝑋”⟩⟩)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ⟨cop 4586  ‘cfv 6492  ⟨“cs1 14521  mVRcmvar 35657  mTypecmty 35658  mVHcmvh 35668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-s1 14522  df-mvh 35688

This theorem is referenced by:  mvhf1  35755  msubvrs  35756