Theorem mvhval 33736

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 6819	. . 3 ⊢ (𝑣 = 𝑋 → (𝑌‘𝑣) = (𝑌‘𝑋))
2		s1eq 14396	. . 3 ⊢ (𝑣 = 𝑋 → ⟨“𝑣”⟩ = ⟨“𝑋”⟩)
3	1, 2	opeq12d 4824	. 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 33735	. 2 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
8		opex 5403	. 2 ⊢ ⟨(𝑌‘𝑋), ⟨“𝑋”⟩⟩ ∈ V
9	3, 7, 8	fvmpt 6925	1 ⊢ (𝑋 ∈ 𝑉 → (𝐻‘𝑋) = ⟨(𝑌‘𝑋), ⟨“𝑋”⟩⟩)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ⟨cop 4578  ‘cfv 6473  ⟨“cs1 14391  mVRcmvar 33663  mTypecmty 33664  mVHcmvh 33674

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pr 5369

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-s1 14392  df-mvh 33694

This theorem is referenced by:  mvhf1  33761  msubvrs  33762