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Theorem msubfval 35033
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 35032 . . . . 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 766 . . . . . . . 8 ((((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) β†’ 𝑓 = 𝐹)
98fveq2d 6886 . . . . . . 7 ((((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) β†’ (π‘‚β€˜π‘“) = (π‘‚β€˜πΉ))
109fveq1d 6884 . . . . . 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 4873 . . . . 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 5239 . . . 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 6896 . . . . . . 7 𝑅 ∈ V
141fvexi 6896 . . . . . . 7 𝑉 ∈ V
1513, 14pm3.2i 470 . . . . . 6 (𝑅 ∈ V ∧ 𝑉 ∈ V)
1615a1i 11 . . . . 5 (𝑇 ∈ V β†’ (𝑅 ∈ V ∧ 𝑉 ∈ V))
17 elpm2r 8836 . . . . 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 6896 . . . . . 6 𝐸 ∈ V
2019mptex 7217 . . . . 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 6996 . . 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 6926 . . . . 5 (βˆ…β€˜πΉ) = βˆ…
25 mpt0 6683 . . . . 5 (𝑒 ∈ βˆ… ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩) = βˆ…
2624, 25eqtr4i 2755 . . . 4 (βˆ…β€˜πΉ) = (𝑒 ∈ βˆ… ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩)
27 fvprc 6874 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mSubstβ€˜π‘‡) = βˆ…)
283, 27eqtrid 2776 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝑆 = βˆ…)
2928fveq1d 6884 . . . 4 (Β¬ 𝑇 ∈ V β†’ (π‘†β€˜πΉ) = (βˆ…β€˜πΉ))
30 fvprc 6874 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mExβ€˜π‘‡) = βˆ…)
314, 30eqtrid 2776 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝐸 = βˆ…)
3231mpteq1d 5234 . . . 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 2790 . . 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 1533   ∈ wcel 2098  Vcvv 3466   βŠ† wss 3941  βˆ…c0 4315  βŸ¨cop 4627   ↦ cmpt 5222  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402  1st c1st 7967  2nd c2nd 7968   ↑pm cpm 8818  mVRcmvar 34970  mRExcmrex 34975  mExcmex 34976  mRSubstcmrsub 34979  mSubstcmsub 34980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-pm 8820  df-msub 35000
This theorem is referenced by:  msubval  35034  msubrn  35038
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