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Theorem msubffval 32777
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 3489 . 2 (𝑇𝑊𝑇 ∈ V)
2 msubffval.s . . 3 𝑆 = (mSubst‘𝑇)
3 fveq2 6643 . . . . . . 7 (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
4 msubffval.r . . . . . . 7 𝑅 = (mREx‘𝑇)
53, 4syl6eqr 2874 . . . . . 6 (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
6 fveq2 6643 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
7 msubffval.v . . . . . . 7 𝑉 = (mVR‘𝑇)
86, 7syl6eqr 2874 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
95, 8oveq12d 7148 . . . . 5 (𝑡 = 𝑇 → ((mREx‘𝑡) ↑pm (mVR‘𝑡)) = (𝑅pm 𝑉))
10 fveq2 6643 . . . . . . 7 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
11 msubffval.e . . . . . . 7 𝐸 = (mEx‘𝑇)
1210, 11syl6eqr 2874 . . . . . 6 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
13 fveq2 6643 . . . . . . . . . 10 (𝑡 = 𝑇 → (mRSubst‘𝑡) = (mRSubst‘𝑇))
14 msubffval.o . . . . . . . . . 10 𝑂 = (mRSubst‘𝑇)
1513, 14syl6eqr 2874 . . . . . . . . 9 (𝑡 = 𝑇 → (mRSubst‘𝑡) = 𝑂)
1615fveq1d 6645 . . . . . . . 8 (𝑡 = 𝑇 → ((mRSubst‘𝑡)‘𝑓) = (𝑂𝑓))
1716fveq1d 6645 . . . . . . 7 (𝑡 = 𝑇 → (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒)) = ((𝑂𝑓)‘(2nd𝑒)))
1817opeq2d 4783 . . . . . 6 (𝑡 = 𝑇 → ⟨(1st𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒))⟩ = ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)
1912, 18mpteq12dv 5124 . . . . 5 (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩))
209, 19mpteq12dv 5124 . . . 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 32745 . . . 4 mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒))⟩)))
22 ovex 7163 . . . . 5 (𝑅pm 𝑉) ∈ V
2322mptex 6959 . . . 4 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)) ∈ V
2420, 21, 23fvmpt 6741 . . 3 (𝑇 ∈ V → (mSubst‘𝑇) = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
252, 24syl5eq 2868 . 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 1538  wcel 2115  Vcvv 3471  cop 4546  cmpt 5119  cfv 6328  (class class class)co 7130  1st c1st 7662  2nd c2nd 7663  pm cpm 8382  mVRcmvar 32715  mRExcmrex 32720  mExcmex 32721  mRSubstcmrsub 32724  mSubstcmsub 32725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-msub 32745
This theorem is referenced by:  msubfval  32778  elmsubrn  32782  msubrn  32783  msubff  32784
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