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Theorem mvhval 30685
 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 (𝑋𝑉 → (𝐻𝑋) = ⟨(𝑌𝑋), ⟨“𝑋”⟩⟩)

Proof of Theorem mvhval
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . 3 (𝑣 = 𝑋 → (𝑌𝑣) = (𝑌𝑋))
2 s1eq 13233 . . 3 (𝑣 = 𝑋 → ⟨“𝑣”⟩ = ⟨“𝑋”⟩)
31, 2opeq12d 4348 . 2 (𝑣 = 𝑋 → ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩ = ⟨(𝑌𝑋), ⟨“𝑋”⟩⟩)
4 mvhfval.v . . 3 𝑉 = (mVR‘𝑇)
5 mvhfval.y . . 3 𝑌 = (mType‘𝑇)
6 mvhfval.h . . 3 𝐻 = (mVH‘𝑇)
74, 5, 6mvhfval 30684 . 2 𝐻 = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
8 opex 4859 . 2 ⟨(𝑌𝑋), ⟨“𝑋”⟩⟩ ∈ V
93, 7, 8fvmpt 6191 1 (𝑋𝑉 → (𝐻𝑋) = ⟨(𝑌𝑋), ⟨“𝑋”⟩⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ⟨cop 4131  ‘cfv 5804  ⟨“cs1 13149  mVRcmvar 30612  mTypecmty 30613  mVHcmvh 30623 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-s1 13157  df-mvh 30643 This theorem is referenced by:  mvhf1  30710  msubvrs  30711
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