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Theorem mvhval 31769
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 6333 . . 3 (𝑣 = 𝑋 → (𝑌𝑣) = (𝑌𝑋))
2 s1eq 13580 . . 3 (𝑣 = 𝑋 → ⟨“𝑣”⟩ = ⟨“𝑋”⟩)
31, 2opeq12d 4548 . 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 31768 . 2 𝐻 = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)
8 opex 5061 . 2 ⟨(𝑌𝑋), ⟨“𝑋”⟩⟩ ∈ V
93, 7, 8fvmpt 6426 1 (𝑋𝑉 → (𝐻𝑋) = ⟨(𝑌𝑋), ⟨“𝑋”⟩⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cop 4323  cfv 6030  ⟨“cs1 13490  mVRcmvar 31696  mTypecmty 31697  mVHcmvh 31707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-s1 13498  df-mvh 31727
This theorem is referenced by:  mvhf1  31794  msubvrs  31795
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