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Theorem msubfval 33057
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 33056 . . . . 5 (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
76adantr 484 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
8 simplr 769 . . . . . . . 8 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → 𝑓 = 𝐹)
98fveq2d 6678 . . . . . . 7 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → (𝑂𝑓) = (𝑂𝐹))
109fveq1d 6676 . . . . . 6 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → ((𝑂𝑓)‘(2nd𝑒)) = ((𝑂𝐹)‘(2nd𝑒)))
1110opeq2d 4768 . . . . 5 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩ = ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)
1211mpteq2dva 5125 . . . 4 (((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
132fvexi 6688 . . . . . . 7 𝑅 ∈ V
141fvexi 6688 . . . . . . 7 𝑉 ∈ V
1513, 14pm3.2i 474 . . . . . 6 (𝑅 ∈ V ∧ 𝑉 ∈ V)
1615a1i 11 . . . . 5 (𝑇 ∈ V → (𝑅 ∈ V ∧ 𝑉 ∈ V))
17 elpm2r 8455 . . . . 5 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝐹 ∈ (𝑅pm 𝑉))
1816, 17sylan 583 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝐹 ∈ (𝑅pm 𝑉))
194fvexi 6688 . . . . . 6 𝐸 ∈ V
2019mptex 6996 . . . . 5 (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) ∈ V
2120a1i 11 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) ∈ V)
227, 12, 18, 21fvmptd 6782 . . 3 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
2322ex 416 . 2 (𝑇 ∈ V → ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)))
24 0fv 6713 . . . . 5 (∅‘𝐹) = ∅
25 mpt0 6479 . . . . 5 (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) = ∅
2624, 25eqtr4i 2764 . . . 4 (∅‘𝐹) = (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)
27 fvprc 6666 . . . . . 6 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
283, 27syl5eq 2785 . . . . 5 𝑇 ∈ V → 𝑆 = ∅)
2928fveq1d 6676 . . . 4 𝑇 ∈ V → (𝑆𝐹) = (∅‘𝐹))
30 fvprc 6666 . . . . . 6 𝑇 ∈ V → (mEx‘𝑇) = ∅)
314, 30syl5eq 2785 . . . . 5 𝑇 ∈ V → 𝐸 = ∅)
3231mpteq1d 5119 . . . 4 𝑇 ∈ V → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) = (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
3326, 29, 323eqtr4a 2799 . . 3 𝑇 ∈ V → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
3433a1d 25 . 2 𝑇 ∈ V → ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)))
3523, 34pm2.61i 185 1 ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1542  wcel 2114  Vcvv 3398  wss 3843  c0 4211  cop 4522  cmpt 5110  wf 6335  cfv 6339  (class class class)co 7170  1st c1st 7712  2nd c2nd 7713  pm cpm 8438  mVRcmvar 32994  mRExcmrex 32999  mExcmex 33000  mRSubstcmrsub 33003  mSubstcmsub 33004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-pm 8440  df-msub 33024
This theorem is referenced by:  msubval  33058  msubrn  33062
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