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Theorem msubffval 34509
Description: A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msubffval.v 𝑉 = (mVRβ€˜π‘‡)
msubffval.r 𝑅 = (mRExβ€˜π‘‡)
msubffval.s 𝑆 = (mSubstβ€˜π‘‡)
msubffval.e 𝐸 = (mExβ€˜π‘‡)
msubffval.o 𝑂 = (mRSubstβ€˜π‘‡)
Assertion
Ref Expression
msubffval (𝑇 ∈ π‘Š β†’ 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)))
Distinct variable groups:   𝑒,𝑓,𝐸   𝑒,𝑂,𝑓   𝑅,𝑒,𝑓   𝑇,𝑒,𝑓   𝑒,𝑉,𝑓
Allowed substitution hints:   𝑆(𝑒,𝑓)   π‘Š(𝑒,𝑓)

Proof of Theorem msubffval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝑇 ∈ π‘Š β†’ 𝑇 ∈ V)
2 msubffval.s . . 3 𝑆 = (mSubstβ€˜π‘‡)
3 fveq2 6891 . . . . . . 7 (𝑑 = 𝑇 β†’ (mRExβ€˜π‘‘) = (mRExβ€˜π‘‡))
4 msubffval.r . . . . . . 7 𝑅 = (mRExβ€˜π‘‡)
53, 4eqtr4di 2790 . . . . . 6 (𝑑 = 𝑇 β†’ (mRExβ€˜π‘‘) = 𝑅)
6 fveq2 6891 . . . . . . 7 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = (mVRβ€˜π‘‡))
7 msubffval.v . . . . . . 7 𝑉 = (mVRβ€˜π‘‡)
86, 7eqtr4di 2790 . . . . . 6 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = 𝑉)
95, 8oveq12d 7426 . . . . 5 (𝑑 = 𝑇 β†’ ((mRExβ€˜π‘‘) ↑pm (mVRβ€˜π‘‘)) = (𝑅 ↑pm 𝑉))
10 fveq2 6891 . . . . . . 7 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = (mExβ€˜π‘‡))
11 msubffval.e . . . . . . 7 𝐸 = (mExβ€˜π‘‡)
1210, 11eqtr4di 2790 . . . . . 6 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = 𝐸)
13 fveq2 6891 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ (mRSubstβ€˜π‘‘) = (mRSubstβ€˜π‘‡))
14 msubffval.o . . . . . . . . . 10 𝑂 = (mRSubstβ€˜π‘‡)
1513, 14eqtr4di 2790 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (mRSubstβ€˜π‘‘) = 𝑂)
1615fveq1d 6893 . . . . . . . 8 (𝑑 = 𝑇 β†’ ((mRSubstβ€˜π‘‘)β€˜π‘“) = (π‘‚β€˜π‘“))
1716fveq1d 6893 . . . . . . 7 (𝑑 = 𝑇 β†’ (((mRSubstβ€˜π‘‘)β€˜π‘“)β€˜(2nd β€˜π‘’)) = ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’)))
1817opeq2d 4880 . . . . . 6 (𝑑 = 𝑇 β†’ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‘)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩ = ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)
1912, 18mpteq12dv 5239 . . . . 5 (𝑑 = 𝑇 β†’ (𝑒 ∈ (mExβ€˜π‘‘) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‘)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩))
209, 19mpteq12dv 5239 . . . 4 (𝑑 = 𝑇 β†’ (𝑓 ∈ ((mRExβ€˜π‘‘) ↑pm (mVRβ€˜π‘‘)) ↦ (𝑒 ∈ (mExβ€˜π‘‘) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‘)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)))
21 df-msub 34477 . . . 4 mSubst = (𝑑 ∈ V ↦ (𝑓 ∈ ((mRExβ€˜π‘‘) ↑pm (mVRβ€˜π‘‘)) ↦ (𝑒 ∈ (mExβ€˜π‘‘) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‘)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)))
22 ovex 7441 . . . . 5 (𝑅 ↑pm 𝑉) ∈ V
2322mptex 7224 . . . 4 (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)) ∈ V
2420, 21, 23fvmpt 6998 . . 3 (𝑇 ∈ V β†’ (mSubstβ€˜π‘‡) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)))
252, 24eqtrid 2784 . 2 (𝑇 ∈ V β†’ 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)))
261, 25syl 17 1 (𝑇 ∈ π‘Š β†’ 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3474  βŸ¨cop 4634   ↦ cmpt 5231  β€˜cfv 6543  (class class class)co 7408  1st c1st 7972  2nd c2nd 7973   ↑pm cpm 8820  mVRcmvar 34447  mRExcmrex 34452  mExcmex 34453  mRSubstcmrsub 34456  mSubstcmsub 34457
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-msub 34477
This theorem is referenced by:  msubfval  34510  elmsubrn  34514  msubrn  34515  msubff  34516
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