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Theorem vrgpfval 19799
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 3499 . . 3 (𝐼𝑉𝐼 ∈ V)
3 id 22 . . . . 5 (𝑖 = 𝐼𝑖 = 𝐼)
4 fveq2 6907 . . . . . . 7 (𝑖 = 𝐼 → ( ~FG𝑖) = ( ~FG𝐼))
5 vrgpfval.r . . . . . . 7 = ( ~FG𝐼)
64, 5eqtr4di 2793 . . . . . 6 (𝑖 = 𝐼 → ( ~FG𝑖) = )
76eceq2d 8787 . . . . 5 (𝑖 = 𝐼 → [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖) = [⟨“⟨𝑗, ∅⟩”⟩] )
83, 7mpteq12dv 5239 . . . 4 (𝑖 = 𝐼 → (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
9 df-vrgp 19744 . . . 4 varFGrp = (𝑖 ∈ V ↦ (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)))
10 vex 3482 . . . . 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 2787 1 (𝐼𝑉𝑈 = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  cop 4637  cmpt 5231  cfv 6563  [cec 8742  ⟨“cs1 14630   ~FG cefg 19739  varFGrpcvrgp 19741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ec 8746  df-vrgp 19744
This theorem is referenced by:  vrgpval  19800  vrgpf  19801
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