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Theorem msubfval 32657
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 32656 . . . . 5 (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
76adantr 481 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
8 simplr 765 . . . . . . . 8 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → 𝑓 = 𝐹)
98fveq2d 6670 . . . . . . 7 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → (𝑂𝑓) = (𝑂𝐹))
109fveq1d 6668 . . . . . 6 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → ((𝑂𝑓)‘(2nd𝑒)) = ((𝑂𝐹)‘(2nd𝑒)))
1110opeq2d 4808 . . . . 5 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩ = ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)
1211mpteq2dva 5157 . . . 4 (((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
132fvexi 6680 . . . . . . 7 𝑅 ∈ V
141fvexi 6680 . . . . . . 7 𝑉 ∈ V
1513, 14pm3.2i 471 . . . . . 6 (𝑅 ∈ V ∧ 𝑉 ∈ V)
1615a1i 11 . . . . 5 (𝑇 ∈ V → (𝑅 ∈ V ∧ 𝑉 ∈ V))
17 elpm2r 8417 . . . . 5 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝐹 ∈ (𝑅pm 𝑉))
1816, 17sylan 580 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝐹 ∈ (𝑅pm 𝑉))
194fvexi 6680 . . . . . 6 𝐸 ∈ V
2019mptex 6984 . . . . 5 (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) ∈ V
2120a1i 11 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) ∈ V)
227, 12, 18, 21fvmptd 6770 . . 3 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
2322ex 413 . 2 (𝑇 ∈ V → ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)))
24 0fv 6705 . . . . 5 (∅‘𝐹) = ∅
25 mpt0 6486 . . . . 5 (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) = ∅
2624, 25eqtr4i 2851 . . . 4 (∅‘𝐹) = (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)
27 fvprc 6659 . . . . . 6 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
283, 27syl5eq 2872 . . . . 5 𝑇 ∈ V → 𝑆 = ∅)
2928fveq1d 6668 . . . 4 𝑇 ∈ V → (𝑆𝐹) = (∅‘𝐹))
30 fvprc 6659 . . . . . 6 𝑇 ∈ V → (mEx‘𝑇) = ∅)
314, 30syl5eq 2872 . . . . 5 𝑇 ∈ V → 𝐸 = ∅)
3231mpteq1d 5151 . . . 4 𝑇 ∈ V → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) = (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
3326, 29, 323eqtr4a 2886 . . 3 𝑇 ∈ V → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
3433a1d 25 . 2 𝑇 ∈ V → ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)))
3523, 34pm2.61i 183 1 ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1530  wcel 2107  Vcvv 3499  wss 3939  c0 4294  cop 4569  cmpt 5142  wf 6347  cfv 6351  (class class class)co 7151  1st c1st 7681  2nd c2nd 7682  pm cpm 8400  mVRcmvar 32594  mRExcmrex 32599  mExcmex 32600  mRSubstcmrsub 32603  mSubstcmsub 32604
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-pm 8402  df-msub 32624
This theorem is referenced by:  msubval  32658  msubrn  32662
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