Theorem mvhval 35494

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 6840	. . 3 ⊢ (𝑣 = 𝑋 → (𝑌‘𝑣) = (𝑌‘𝑋))
2		s1eq 14541	. . 3 ⊢ (𝑣 = 𝑋 → ⟨“𝑣”⟩ = ⟨“𝑋”⟩)
3	1, 2	opeq12d 4841	. 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 35493	. 2 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
8		opex 5419	. 2 ⊢ ⟨(𝑌‘𝑋), ⟨“𝑋”⟩⟩ ∈ V
9	3, 7, 8	fvmpt 6950	1 ⊢ (𝑋 ∈ 𝑉 → (𝐻‘𝑋) = ⟨(𝑌‘𝑋), ⟨“𝑋”⟩⟩)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4591  ‘cfv 6499  ⟨“cs1 14536  mVRcmvar 35421  mTypecmty 35422  mVHcmvh 35432

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-s1 14537  df-mvh 35452

This theorem is referenced by:  mvhf1  35519  msubvrs  35520