Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  msubffval Structured version   Visualization version   GIF version

Theorem msubffval 34452
Description: A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msubffval.v 𝑉 = (mVR‘𝑇)
msubffval.r 𝑅 = (mREx‘𝑇)
msubffval.s 𝑆 = (mSubst‘𝑇)
msubffval.e 𝐸 = (mEx‘𝑇)
msubffval.o 𝑂 = (mRSubst‘𝑇)
Assertion
Ref Expression
msubffval (𝑇𝑊𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
Distinct variable groups:   𝑒,𝑓,𝐸   𝑒,𝑂,𝑓   𝑅,𝑒,𝑓   𝑇,𝑒,𝑓   𝑒,𝑉,𝑓
Allowed substitution hints:   𝑆(𝑒,𝑓)   𝑊(𝑒,𝑓)

Proof of Theorem msubffval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 elex 3493 . 2 (𝑇𝑊𝑇 ∈ V)
2 msubffval.s . . 3 𝑆 = (mSubst‘𝑇)
3 fveq2 6888 . . . . . . 7 (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
4 msubffval.r . . . . . . 7 𝑅 = (mREx‘𝑇)
53, 4eqtr4di 2791 . . . . . 6 (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
6 fveq2 6888 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
7 msubffval.v . . . . . . 7 𝑉 = (mVR‘𝑇)
86, 7eqtr4di 2791 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
95, 8oveq12d 7422 . . . . 5 (𝑡 = 𝑇 → ((mREx‘𝑡) ↑pm (mVR‘𝑡)) = (𝑅pm 𝑉))
10 fveq2 6888 . . . . . . 7 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
11 msubffval.e . . . . . . 7 𝐸 = (mEx‘𝑇)
1210, 11eqtr4di 2791 . . . . . 6 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
13 fveq2 6888 . . . . . . . . . 10 (𝑡 = 𝑇 → (mRSubst‘𝑡) = (mRSubst‘𝑇))
14 msubffval.o . . . . . . . . . 10 𝑂 = (mRSubst‘𝑇)
1513, 14eqtr4di 2791 . . . . . . . . 9 (𝑡 = 𝑇 → (mRSubst‘𝑡) = 𝑂)
1615fveq1d 6890 . . . . . . . 8 (𝑡 = 𝑇 → ((mRSubst‘𝑡)‘𝑓) = (𝑂𝑓))
1716fveq1d 6890 . . . . . . 7 (𝑡 = 𝑇 → (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒)) = ((𝑂𝑓)‘(2nd𝑒)))
1817opeq2d 4879 . . . . . 6 (𝑡 = 𝑇 → ⟨(1st𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒))⟩ = ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)
1912, 18mpteq12dv 5238 . . . . 5 (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩))
209, 19mpteq12dv 5238 . . . 4 (𝑡 = 𝑇 → (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒))⟩)) = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
21 df-msub 34420 . . . 4 mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒))⟩)))
22 ovex 7437 . . . . 5 (𝑅pm 𝑉) ∈ V
2322mptex 7220 . . . 4 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)) ∈ V
2420, 21, 23fvmpt 6994 . . 3 (𝑇 ∈ V → (mSubst‘𝑇) = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
252, 24eqtrid 2785 . 2 (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
261, 25syl 17 1 (𝑇𝑊𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  cop 4633  cmpt 5230  cfv 6540  (class class class)co 7404  1st c1st 7968  2nd c2nd 7969  pm cpm 8817  mVRcmvar 34390  mRExcmrex 34395  mExcmex 34396  mRSubstcmrsub 34399  mSubstcmsub 34400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-msub 34420
This theorem is referenced by:  msubfval  34453  elmsubrn  34457  msubrn  34458  msubff  34459
  Copyright terms: Public domain W3C validator