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Theorem mvhval 31759
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 6353 . . 3 (𝑣 = 𝑋 → (𝑌𝑣) = (𝑌𝑋))
2 s1eq 13590 . . 3 (𝑣 = 𝑋 → ⟨“𝑣”⟩ = ⟨“𝑋”⟩)
31, 2opeq12d 4561 . 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 31758 . 2 𝐻 = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
8 opex 5081 . 2 ⟨(𝑌𝑋), ⟨“𝑋”⟩⟩ ∈ V
93, 7, 8fvmpt 6445 1 (𝑋𝑉 → (𝐻𝑋) = ⟨(𝑌𝑋), ⟨“𝑋”⟩⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  cop 4327  cfv 6049  ⟨“cs1 13500  mVRcmvar 31686  mTypecmty 31687  mVHcmvh 31697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-s1 13508  df-mvh 31717
This theorem is referenced by:  mvhf1  31784  msubvrs  31785
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