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Theorem mrsubfval 31166
Description: The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mrsubffval.c 𝐶 = (mCN‘𝑇)
mrsubffval.v 𝑉 = (mVR‘𝑇)
mrsubffval.r 𝑅 = (mREx‘𝑇)
mrsubffval.s 𝑆 = (mRSubst‘𝑇)
mrsubffval.g 𝐺 = (freeMnd‘(𝐶𝑉))
Assertion
Ref Expression
mrsubfval ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
Distinct variable groups:   𝑣,𝑒,𝐴   𝐶,𝑒,𝑣   𝑒,𝐹,𝑣   𝑅,𝑒,𝑣   𝑒,𝐺   𝑇,𝑒,𝑣   𝑒,𝑉,𝑣
Allowed substitution hints:   𝑆(𝑣,𝑒)   𝐺(𝑣)

Proof of Theorem mrsubfval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 mrsubffval.c . . . . . 6 𝐶 = (mCN‘𝑇)
2 mrsubffval.v . . . . . 6 𝑉 = (mVR‘𝑇)
3 mrsubffval.r . . . . . 6 𝑅 = (mREx‘𝑇)
4 mrsubffval.s . . . . . 6 𝑆 = (mRSubst‘𝑇)
5 mrsubffval.g . . . . . 6 𝐺 = (freeMnd‘(𝐶𝑉))
61, 2, 3, 4, 5mrsubffval 31165 . . . . 5 (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
76adantr 481 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
8 dmeq 5294 . . . . . . . . . . 11 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
9 fdm 6018 . . . . . . . . . . . 12 (𝐹:𝐴𝑅 → dom 𝐹 = 𝐴)
109ad2antrl 763 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → dom 𝐹 = 𝐴)
118, 10sylan9eqr 2677 . . . . . . . . . 10 (((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) → dom 𝑓 = 𝐴)
1211eleq2d 2684 . . . . . . . . 9 (((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) → (𝑣 ∈ dom 𝑓𝑣𝐴))
13 simpr 477 . . . . . . . . . 10 (((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
1413fveq1d 6160 . . . . . . . . 9 (((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) → (𝑓𝑣) = (𝐹𝑣))
1512, 14ifbieq1d 4087 . . . . . . . 8 (((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) → if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩) = if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩))
1615mpteq2dv 4715 . . . . . . 7 (((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) → (𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)))
1716coeq1d 5253 . . . . . 6 (((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) → ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) = ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))
1817oveq2d 6631 . . . . 5 (((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) → (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) = (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))
1918mpteq2dv 4715 . . . 4 (((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) ∧ 𝑓 = 𝐹) → (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
20 fvex 6168 . . . . . . 7 (mREx‘𝑇) ∈ V
213, 20eqeltri 2694 . . . . . 6 𝑅 ∈ V
2221a1i 11 . . . . 5 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝑅 ∈ V)
23 fvex 6168 . . . . . . 7 (mVR‘𝑇) ∈ V
242, 23eqeltri 2694 . . . . . 6 𝑉 ∈ V
2524a1i 11 . . . . 5 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝑉 ∈ V)
26 simprl 793 . . . . 5 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝐹:𝐴𝑅)
27 simprr 795 . . . . 5 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝐴𝑉)
28 elpm2r 7835 . . . . 5 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝐹 ∈ (𝑅pm 𝑉))
2922, 25, 26, 27, 28syl22anc 1324 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → 𝐹 ∈ (𝑅pm 𝑉))
3021mptex 6451 . . . . 5 (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) ∈ V
3130a1i 11 . . . 4 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) ∈ V)
327, 19, 29, 31fvmptd 6255 . . 3 ((𝑇 ∈ V ∧ (𝐹:𝐴𝑅𝐴𝑉)) → (𝑆𝐹) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
3332ex 450 . 2 (𝑇 ∈ V → ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
34 0fv 6194 . . . 4 (∅‘𝐹) = ∅
35 fvprc 6152 . . . . . 6 𝑇 ∈ V → (mRSubst‘𝑇) = ∅)
364, 35syl5eq 2667 . . . . 5 𝑇 ∈ V → 𝑆 = ∅)
3736fveq1d 6160 . . . 4 𝑇 ∈ V → (𝑆𝐹) = (∅‘𝐹))
38 fvprc 6152 . . . . . . 7 𝑇 ∈ V → (mREx‘𝑇) = ∅)
393, 38syl5eq 2667 . . . . . 6 𝑇 ∈ V → 𝑅 = ∅)
4039mpteq1d 4708 . . . . 5 𝑇 ∈ V → (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) = (𝑒 ∈ ∅ ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
41 mpt0 5988 . . . . 5 (𝑒 ∈ ∅ ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) = ∅
4240, 41syl6eq 2671 . . . 4 𝑇 ∈ V → (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) = ∅)
4334, 37, 423eqtr4a 2681 . . 3 𝑇 ∈ V → (𝑆𝐹) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
4443a1d 25 . 2 𝑇 ∈ V → ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
4533, 44pm2.61i 176 1 ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  cun 3558  wss 3560  c0 3897  ifcif 4064  cmpt 4683  dom cdm 5084  ccom 5088  wf 5853  cfv 5857  (class class class)co 6615  pm cpm 7818  ⟨“cs1 13249   Σg cgsu 16041  freeMndcfrmd 17324  mCNcmcn 31118  mVRcmvar 31119  mRExcmrex 31124  mRSubstcmrsub 31128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-pm 7820  df-mrsub 31148
This theorem is referenced by:  mrsubval  31167  mrsubrn  31171  elmrsubrn  31178
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