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Theorem msubfval 35134
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
msubfval ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩))
Distinct variable groups:   𝑒,𝐸   𝑒,𝑂   𝑅,𝑒   𝑇,𝑒   𝑒,𝑉   𝐴,𝑒   𝑒,𝐹
Allowed substitution hint:   𝑆(𝑒)

Proof of Theorem msubfval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 msubffval.v . . . . . 6 𝑉 = (mVRβ€˜π‘‡)
2 msubffval.r . . . . . 6 𝑅 = (mRExβ€˜π‘‡)
3 msubffval.s . . . . . 6 𝑆 = (mSubstβ€˜π‘‡)
4 msubffval.e . . . . . 6 𝐸 = (mExβ€˜π‘‡)
5 msubffval.o . . . . . 6 𝑂 = (mRSubstβ€˜π‘‡)
61, 2, 3, 4, 5msubffval 35133 . . . . 5 (𝑇 ∈ V β†’ 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)))
76adantr 480 . . . 4 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)))
8 simplr 768 . . . . . . . 8 ((((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) β†’ 𝑓 = 𝐹)
98fveq2d 6901 . . . . . . 7 ((((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) β†’ (π‘‚β€˜π‘“) = (π‘‚β€˜πΉ))
109fveq1d 6899 . . . . . 6 ((((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) β†’ ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’)) = ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’)))
1110opeq2d 4881 . . . . 5 ((((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) β†’ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩ = ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩)
1211mpteq2dva 5248 . . . 4 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩))
132fvexi 6911 . . . . . . 7 𝑅 ∈ V
141fvexi 6911 . . . . . . 7 𝑉 ∈ V
1513, 14pm3.2i 470 . . . . . 6 (𝑅 ∈ V ∧ 𝑉 ∈ V)
1615a1i 11 . . . . 5 (𝑇 ∈ V β†’ (𝑅 ∈ V ∧ 𝑉 ∈ V))
17 elpm2r 8864 . . . . 5 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝐹 ∈ (𝑅 ↑pm 𝑉))
1816, 17sylan 579 . . . 4 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝐹 ∈ (𝑅 ↑pm 𝑉))
194fvexi 6911 . . . . . 6 𝐸 ∈ V
2019mptex 7235 . . . . 5 (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩) ∈ V
2120a1i 11 . . . 4 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩) ∈ V)
227, 12, 18, 21fvmptd 7012 . . 3 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩))
2322ex 412 . 2 (𝑇 ∈ V β†’ ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩)))
24 0fv 6941 . . . . 5 (βˆ…β€˜πΉ) = βˆ…
25 mpt0 6697 . . . . 5 (𝑒 ∈ βˆ… ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩) = βˆ…
2624, 25eqtr4i 2759 . . . 4 (βˆ…β€˜πΉ) = (𝑒 ∈ βˆ… ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩)
27 fvprc 6889 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mSubstβ€˜π‘‡) = βˆ…)
283, 27eqtrid 2780 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝑆 = βˆ…)
2928fveq1d 6899 . . . 4 (Β¬ 𝑇 ∈ V β†’ (π‘†β€˜πΉ) = (βˆ…β€˜πΉ))
30 fvprc 6889 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mExβ€˜π‘‡) = βˆ…)
314, 30eqtrid 2780 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝐸 = βˆ…)
3231mpteq1d 5243 . . . 4 (Β¬ 𝑇 ∈ V β†’ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩) = (𝑒 ∈ βˆ… ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩))
3326, 29, 323eqtr4a 2794 . . 3 (Β¬ 𝑇 ∈ V β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩))
3433a1d 25 . 2 (Β¬ 𝑇 ∈ V β†’ ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩)))
3523, 34pm2.61i 182 1 ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3471   βŠ† wss 3947  βˆ…c0 4323  βŸ¨cop 4635   ↦ cmpt 5231  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420  1st c1st 7991  2nd c2nd 7992   ↑pm cpm 8846  mVRcmvar 35071  mRExcmrex 35076  mExcmex 35077  mRSubstcmrsub 35080  mSubstcmsub 35081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-pm 8848  df-msub 35101
This theorem is referenced by:  msubval  35135  msubrn  35139
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