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Theorem mvhval 32776
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 6665 . . 3 (𝑣 = 𝑋 → (𝑌𝑣) = (𝑌𝑋))
2 s1eq 13948 . . 3 (𝑣 = 𝑋 → ⟨“𝑣”⟩ = ⟨“𝑋”⟩)
31, 2opeq12d 4805 . 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 32775 . 2 𝐻 = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
8 opex 5349 . 2 ⟨(𝑌𝑋), ⟨“𝑋”⟩⟩ ∈ V
93, 7, 8fvmpt 6763 1 (𝑋𝑉 → (𝐻𝑋) = ⟨(𝑌𝑋), ⟨“𝑋”⟩⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2110  cop 4567  cfv 6350  ⟨“cs1 13943  mVRcmvar 32703  mTypecmty 32704  mVHcmvh 32714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-s1 13944  df-mvh 32734
This theorem is referenced by:  mvhf1  32801  msubvrs  32802
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