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Theorem msubffval 34181
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 3465 . 2 (𝑇 ∈ π‘Š β†’ 𝑇 ∈ V)
2 msubffval.s . . 3 𝑆 = (mSubstβ€˜π‘‡)
3 fveq2 6846 . . . . . . 7 (𝑑 = 𝑇 β†’ (mRExβ€˜π‘‘) = (mRExβ€˜π‘‡))
4 msubffval.r . . . . . . 7 𝑅 = (mRExβ€˜π‘‡)
53, 4eqtr4di 2791 . . . . . 6 (𝑑 = 𝑇 β†’ (mRExβ€˜π‘‘) = 𝑅)
6 fveq2 6846 . . . . . . 7 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = (mVRβ€˜π‘‡))
7 msubffval.v . . . . . . 7 𝑉 = (mVRβ€˜π‘‡)
86, 7eqtr4di 2791 . . . . . 6 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = 𝑉)
95, 8oveq12d 7379 . . . . 5 (𝑑 = 𝑇 β†’ ((mRExβ€˜π‘‘) ↑pm (mVRβ€˜π‘‘)) = (𝑅 ↑pm 𝑉))
10 fveq2 6846 . . . . . . 7 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = (mExβ€˜π‘‡))
11 msubffval.e . . . . . . 7 𝐸 = (mExβ€˜π‘‡)
1210, 11eqtr4di 2791 . . . . . 6 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = 𝐸)
13 fveq2 6846 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ (mRSubstβ€˜π‘‘) = (mRSubstβ€˜π‘‡))
14 msubffval.o . . . . . . . . . 10 𝑂 = (mRSubstβ€˜π‘‡)
1513, 14eqtr4di 2791 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (mRSubstβ€˜π‘‘) = 𝑂)
1615fveq1d 6848 . . . . . . . 8 (𝑑 = 𝑇 β†’ ((mRSubstβ€˜π‘‘)β€˜π‘“) = (π‘‚β€˜π‘“))
1716fveq1d 6848 . . . . . . 7 (𝑑 = 𝑇 β†’ (((mRSubstβ€˜π‘‘)β€˜π‘“)β€˜(2nd β€˜π‘’)) = ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’)))
1817opeq2d 4841 . . . . . 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 5200 . . . . 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 5200 . . . 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 34149 . . . 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 7394 . . . . 5 (𝑅 ↑pm 𝑉) ∈ V
2322mptex 7177 . . . 4 (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)) ∈ V
2420, 21, 23fvmpt 6952 . . 3 (𝑇 ∈ V β†’ (mSubstβ€˜π‘‡) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)))
252, 24eqtrid 2785 . 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 1542   ∈ wcel 2107  Vcvv 3447  βŸ¨cop 4596   ↦ cmpt 5192  β€˜cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924   ↑pm cpm 8772  mVRcmvar 34119  mRExcmrex 34124  mExcmex 34125  mRSubstcmrsub 34128  mSubstcmsub 34129
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 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-msub 34149
This theorem is referenced by:  msubfval  34182  elmsubrn  34186  msubrn  34187  msubff  34188
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