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Theorem msubfval 31750
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 31749 . . . . 5 (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
76adantr 472 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))
8 simplr 809 . . . . . . . 8 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → 𝑓 = 𝐹)
98fveq2d 6358 . . . . . . 7 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → (𝑂𝑓) = (𝑂𝐹))
109fveq1d 6356 . . . . . 6 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → ((𝑂𝑓)‘(2nd𝑒)) = ((𝑂𝐹)‘(2nd𝑒)))
1110opeq2d 4561 . . . . 5 ((((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒𝐸) → ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩ = ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)
1211mpteq2dva 4897 . . . 4 (((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
13 fvex 6364 . . . . . . . 8 (mREx‘𝑇) ∈ V
142, 13eqeltri 2836 . . . . . . 7 𝑅 ∈ V
15 fvex 6364 . . . . . . . 8 (mVR‘𝑇) ∈ V
161, 15eqeltri 2836 . . . . . . 7 𝑉 ∈ V
1714, 16pm3.2i 470 . . . . . 6 (𝑅 ∈ V ∧ 𝑉 ∈ V)
1817a1i 11 . . . . 5 (𝑇 ∈ V → (𝑅 ∈ V ∧ 𝑉 ∈ V))
19 elpm2r 8044 . . . . 5 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝐹 ∈ (𝑅pm 𝑉))
2018, 19sylan 489 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝐹 ∈ (𝑅pm 𝑉))
21 fvex 6364 . . . . . . 7 (mEx‘𝑇) ∈ V
224, 21eqeltri 2836 . . . . . 6 𝐸 ∈ V
2322mptex 6652 . . . . 5 (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) ∈ V
2423a1i 11 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) ∈ V)
257, 12, 20, 24fvmptd 6452 . . 3 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
2625ex 449 . 2 (𝑇 ∈ V → ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)))
27 0fv 6390 . . . . 5 (∅‘𝐹) = ∅
28 mpt0 6183 . . . . 5 (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) = ∅
2927, 28eqtr4i 2786 . . . 4 (∅‘𝐹) = (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)
30 fvprc 6348 . . . . . 6 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
313, 30syl5eq 2807 . . . . 5 𝑇 ∈ V → 𝑆 = ∅)
3231fveq1d 6356 . . . 4 𝑇 ∈ V → (𝑆𝐹) = (∅‘𝐹))
33 fvprc 6348 . . . . . 6 𝑇 ∈ V → (mEx‘𝑇) = ∅)
344, 33syl5eq 2807 . . . . 5 𝑇 ∈ V → 𝐸 = ∅)
3534mpteq1d 4891 . . . 4 𝑇 ∈ V → (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩) = (𝑒 ∈ ∅ ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
3629, 32, 353eqtr4a 2821 . . 3 𝑇 ∈ V → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
3736a1d 25 . 2 𝑇 ∈ V → ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩)))
3826, 37pm2.61i 176 1 ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2140  Vcvv 3341  wss 3716  c0 4059  cop 4328  cmpt 4882  wf 6046  cfv 6050  (class class class)co 6815  1st c1st 7333  2nd c2nd 7334  pm cpm 8027  mVRcmvar 31687  mRExcmrex 31692  mExcmex 31693  mRSubstcmrsub 31696  mSubstcmsub 31697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-pm 8029  df-msub 31717
This theorem is referenced by:  msubval  31751  msubrn  31755
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