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Theorem vrgpfval 19784
Description: The canonical injection from the generating set 𝐼 to the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
vrgpfval.r = ( ~FG𝐼)
vrgpfval.u 𝑈 = (varFGrp𝐼)
Assertion
Ref Expression
vrgpfval (𝐼𝑉𝑈 = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
Distinct variable groups:   𝑗,𝐼   ,𝑗   𝑗,𝑉
Allowed substitution hint:   𝑈(𝑗)

Proof of Theorem vrgpfval
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 vrgpfval.u . 2 𝑈 = (varFGrp𝐼)
2 elex 3501 . . 3 (𝐼𝑉𝐼 ∈ V)
3 id 22 . . . . 5 (𝑖 = 𝐼𝑖 = 𝐼)
4 fveq2 6906 . . . . . . 7 (𝑖 = 𝐼 → ( ~FG𝑖) = ( ~FG𝐼))
5 vrgpfval.r . . . . . . 7 = ( ~FG𝐼)
64, 5eqtr4di 2795 . . . . . 6 (𝑖 = 𝐼 → ( ~FG𝑖) = )
76eceq2d 8788 . . . . 5 (𝑖 = 𝐼 → [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖) = [⟨“⟨𝑗, ∅⟩”⟩] )
83, 7mpteq12dv 5233 . . . 4 (𝑖 = 𝐼 → (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
9 df-vrgp 19729 . . . 4 varFGrp = (𝑖 ∈ V ↦ (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)))
10 vex 3484 . . . . 5 𝑖 ∈ V
1110mptex 7243 . . . 4 (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)) ∈ V
128, 9, 11fvmpt3i 7021 . . 3 (𝐼 ∈ V → (varFGrp𝐼) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
132, 12syl 17 . 2 (𝐼𝑉 → (varFGrp𝐼) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
141, 13eqtrid 2789 1 (𝐼𝑉𝑈 = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cop 4632  cmpt 5225  cfv 6561  [cec 8743  ⟨“cs1 14633   ~FG cefg 19724  varFGrpcvrgp 19726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ec 8747  df-vrgp 19729
This theorem is referenced by:  vrgpval  19785  vrgpf  19786
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