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Theorem mrsubval 32781
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
mrsubval ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → ((𝑆𝐹)‘𝑋) = (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))
Distinct variable groups:   𝑣,𝐴   𝑣,𝐶   𝑣,𝐹   𝑣,𝑅   𝑣,𝑋   𝑣,𝑇   𝑣,𝑉
Allowed substitution hints:   𝑆(𝑣)   𝐺(𝑣)

Proof of Theorem mrsubval
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 mrsubffval.c . . . 4 𝐶 = (mCN‘𝑇)
2 mrsubffval.v . . . 4 𝑉 = (mVR‘𝑇)
3 mrsubffval.r . . . 4 𝑅 = (mREx‘𝑇)
4 mrsubffval.s . . . 4 𝑆 = (mRSubst‘𝑇)
5 mrsubffval.g . . . 4 𝐺 = (freeMnd‘(𝐶𝑉))
61, 2, 3, 4, 5mrsubfval 32780 . . 3 ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
763adant3 1129 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → (𝑆𝐹) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
8 simpr 488 . . . 4 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
98coeq2d 5720 . . 3 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) ∧ 𝑒 = 𝑋) → ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) = ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋))
109oveq2d 7161 . 2 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) ∧ 𝑒 = 𝑋) → (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) = (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))
11 simp3 1135 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → 𝑋𝑅)
12 ovexd 7180 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ∈ V)
137, 10, 11, 12fvmptd 6763 1 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → ((𝑆𝐹)‘𝑋) = (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  Vcvv 3480  cun 3917  wss 3919  ifcif 4449  cmpt 5132  ccom 5546  wf 6339  cfv 6343  (class class class)co 7145  ⟨“cs1 13945   Σg cgsu 16710  freeMndcfrmd 18008  mCNcmcn 32732  mVRcmvar 32733  mRExcmrex 32738  mRSubstcmrsub 32742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7148  df-oprab 7149  df-mpo 7150  df-pm 8399  df-mrsub 32762
This theorem is referenced by:  mrsubcv  32782  mrsub0  32788  mrsubccat  32790
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