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Theorem mvhval 32678
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 6663 . . 3 (𝑣 = 𝑋 → (𝑌𝑣) = (𝑌𝑋))
2 s1eq 13942 . . 3 (𝑣 = 𝑋 → ⟨“𝑣”⟩ = ⟨“𝑋”⟩)
31, 2opeq12d 4803 . 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 32677 . 2 𝐻 = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
8 opex 5347 . 2 ⟨(𝑌𝑋), ⟨“𝑋”⟩⟩ ∈ V
93, 7, 8fvmpt 6761 1 (𝑋𝑉 → (𝐻𝑋) = ⟨(𝑌𝑋), ⟨“𝑋”⟩⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cop 4563  cfv 6348  ⟨“cs1 13937  mVRcmvar 32605  mTypecmty 32606  mVHcmvh 32616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-s1 13938  df-mvh 32636
This theorem is referenced by:  mvhf1  32703  msubvrs  32704
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