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Theorem msubfval 32668
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 32667 . . . . 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 6667 . . . . . . 7 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → (𝑂𝑓) = (𝑂𝐹))
109fveq1d 6665 . . . . . 6 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → ((𝑂𝑓)‘(2nd𝑒)) = ((𝑂𝐹)‘(2nd𝑒)))
1110opeq2d 4802 . . . . 5 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩ = ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)
1211mpteq2dva 5152 . . . 4 (((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
132fvexi 6677 . . . . . . 7 𝑅 ∈ V
141fvexi 6677 . . . . . . 7 𝑉 ∈ V
1513, 14pm3.2i 471 . . . . . 6 (𝑅 ∈ V ∧ 𝑉 ∈ V)
1615a1i 11 . . . . 5 (𝑇 ∈ V → (𝑅 ∈ V ∧ 𝑉 ∈ V))
17 elpm2r 8413 . . . . 5 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝐹 ∈ (𝑅pm 𝑉))
1816, 17sylan 580 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝐹 ∈ (𝑅pm 𝑉))
194fvexi 6677 . . . . . 6 𝐸 ∈ V
2019mptex 6977 . . . . 5 (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) ∈ V
2120a1i 11 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) ∈ V)
227, 12, 18, 21fvmptd 6767 . . 3 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
2322ex 413 . 2 (𝑇 ∈ V → ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)))
24 0fv 6702 . . . . 5 (∅‘𝐹) = ∅
25 mpt0 6483 . . . . 5 (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) = ∅
2624, 25eqtr4i 2844 . . . 4 (∅‘𝐹) = (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)
27 fvprc 6656 . . . . . 6 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
283, 27syl5eq 2865 . . . . 5 𝑇 ∈ V → 𝑆 = ∅)
2928fveq1d 6665 . . . 4 𝑇 ∈ V → (𝑆𝐹) = (∅‘𝐹))
30 fvprc 6656 . . . . . 6 𝑇 ∈ V → (mEx‘𝑇) = ∅)
314, 30syl5eq 2865 . . . . 5 𝑇 ∈ V → 𝐸 = ∅)
3231mpteq1d 5146 . . . 4 𝑇 ∈ V → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) = (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
3326, 29, 323eqtr4a 2879 . . 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 1528  wcel 2105  Vcvv 3492  wss 3933  c0 4288  cop 4563  cmpt 5137  wf 6344  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  pm cpm 8396  mVRcmvar 32605  mRExcmrex 32610  mExcmex 32611  mRSubstcmrsub 32614  mSubstcmsub 32615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-pm 8398  df-msub 32635
This theorem is referenced by:  msubval  32669  msubrn  32673
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