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Theorem msubffval 35528
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 3501 . 2 (𝑇𝑊𝑇 ∈ V)
2 msubffval.s . . 3 𝑆 = (mSubst‘𝑇)
3 fveq2 6906 . . . . . . 7 (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
4 msubffval.r . . . . . . 7 𝑅 = (mREx‘𝑇)
53, 4eqtr4di 2795 . . . . . 6 (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
6 fveq2 6906 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
7 msubffval.v . . . . . . 7 𝑉 = (mVR‘𝑇)
86, 7eqtr4di 2795 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
95, 8oveq12d 7449 . . . . 5 (𝑡 = 𝑇 → ((mREx‘𝑡) ↑pm (mVR‘𝑡)) = (𝑅pm 𝑉))
10 fveq2 6906 . . . . . . 7 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
11 msubffval.e . . . . . . 7 𝐸 = (mEx‘𝑇)
1210, 11eqtr4di 2795 . . . . . 6 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
13 fveq2 6906 . . . . . . . . . 10 (𝑡 = 𝑇 → (mRSubst‘𝑡) = (mRSubst‘𝑇))
14 msubffval.o . . . . . . . . . 10 𝑂 = (mRSubst‘𝑇)
1513, 14eqtr4di 2795 . . . . . . . . 9 (𝑡 = 𝑇 → (mRSubst‘𝑡) = 𝑂)
1615fveq1d 6908 . . . . . . . 8 (𝑡 = 𝑇 → ((mRSubst‘𝑡)‘𝑓) = (𝑂𝑓))
1716fveq1d 6908 . . . . . . 7 (𝑡 = 𝑇 → (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒)) = ((𝑂𝑓)‘(2nd𝑒)))
1817opeq2d 4880 . . . . . 6 (𝑡 = 𝑇 → ⟨(1st𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒))⟩ = ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)
1912, 18mpteq12dv 5233 . . . . 5 (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩))
209, 19mpteq12dv 5233 . . . 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 35496 . . . 4 mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒))⟩)))
22 ovex 7464 . . . . 5 (𝑅pm 𝑉) ∈ V
2322mptex 7243 . . . 4 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)) ∈ V
2420, 21, 23fvmpt 7016 . . 3 (𝑇 ∈ V → (mSubst‘𝑇) = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
252, 24eqtrid 2789 . 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 1540  wcel 2108  Vcvv 3480  cop 4632  cmpt 5225  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  pm cpm 8867  mVRcmvar 35466  mRExcmrex 35471  mExcmex 35472  mRSubstcmrsub 35475  mSubstcmsub 35476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-msub 35496
This theorem is referenced by:  msubfval  35529  elmsubrn  35533  msubrn  35534  msubff  35535
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