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Theorem msubfval 35006
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
msubfval ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
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 = (mVR‘𝑇)
2 msubffval.r . . . . . 6 𝑅 = (mREx‘𝑇)
3 msubffval.s . . . . . 6 𝑆 = (mSubst‘𝑇)
4 msubffval.e . . . . . 6 𝐸 = (mEx‘𝑇)
5 msubffval.o . . . . . 6 𝑂 = (mRSubst‘𝑇)
61, 2, 3, 4, 5msubffval 35005 . . . . 5 (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
76adantr 480 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
8 simplr 766 . . . . . . . 8 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → 𝑓 = 𝐹)
98fveq2d 6886 . . . . . . 7 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → (𝑂𝑓) = (𝑂𝐹))
109fveq1d 6884 . . . . . 6 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → ((𝑂𝑓)‘(2nd𝑒)) = ((𝑂𝐹)‘(2nd𝑒)))
1110opeq2d 4873 . . . . 5 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩ = ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)
1211mpteq2dva 5239 . . . 4 (((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
132fvexi 6896 . . . . . . 7 𝑅 ∈ V
141fvexi 6896 . . . . . . 7 𝑉 ∈ V
1513, 14pm3.2i 470 . . . . . 6 (𝑅 ∈ V ∧ 𝑉 ∈ V)
1615a1i 11 . . . . 5 (𝑇 ∈ V → (𝑅 ∈ V ∧ 𝑉 ∈ V))
17 elpm2r 8836 . . . . 5 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝐹 ∈ (𝑅pm 𝑉))
1816, 17sylan 579 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝐹 ∈ (𝑅pm 𝑉))
194fvexi 6896 . . . . . 6 𝐸 ∈ V
2019mptex 7217 . . . . 5 (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) ∈ V
2120a1i 11 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) ∈ V)
227, 12, 18, 21fvmptd 6996 . . 3 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
2322ex 412 . 2 (𝑇 ∈ V → ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)))
24 0fv 6926 . . . . 5 (∅‘𝐹) = ∅
25 mpt0 6683 . . . . 5 (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) = ∅
2624, 25eqtr4i 2755 . . . 4 (∅‘𝐹) = (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)
27 fvprc 6874 . . . . . 6 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
283, 27eqtrid 2776 . . . . 5 𝑇 ∈ V → 𝑆 = ∅)
2928fveq1d 6884 . . . 4 𝑇 ∈ V → (𝑆𝐹) = (∅‘𝐹))
30 fvprc 6874 . . . . . 6 𝑇 ∈ V → (mEx‘𝑇) = ∅)
314, 30eqtrid 2776 . . . . 5 𝑇 ∈ V → 𝐸 = ∅)
3231mpteq1d 5234 . . . 4 𝑇 ∈ V → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) = (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
3326, 29, 323eqtr4a 2790 . . 3 𝑇 ∈ V → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
3433a1d 25 . 2 𝑇 ∈ V → ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)))
3523, 34pm2.61i 182 1 ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  wss 3941  c0 4315  cop 4627  cmpt 5222  wf 6530  cfv 6534  (class class class)co 7402  1st c1st 7967  2nd c2nd 7968  pm cpm 8818  mVRcmvar 34943  mRExcmrex 34948  mExcmex 34949  mRSubstcmrsub 34952  mSubstcmsub 34953
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 34973
This theorem is referenced by:  msubval  35007  msubrn  35011
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