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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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