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Theorem vrgpfval 18160
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 3207 . . 3 (𝐼𝑉𝐼 ∈ V)
3 id 22 . . . . 5 (𝑖 = 𝐼𝑖 = 𝐼)
4 fveq2 6178 . . . . . . 7 (𝑖 = 𝐼 → ( ~FG𝑖) = ( ~FG𝐼))
5 vrgpfval.r . . . . . . 7 = ( ~FG𝐼)
64, 5syl6eqr 2672 . . . . . 6 (𝑖 = 𝐼 → ( ~FG𝑖) = )
7 eceq2 7769 . . . . . 6 (( ~FG𝑖) = → [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖) = [⟨“⟨𝑗, ∅⟩”⟩] )
86, 7syl 17 . . . . 5 (𝑖 = 𝐼 → [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖) = [⟨“⟨𝑗, ∅⟩”⟩] )
93, 8mpteq12dv 4724 . . . 4 (𝑖 = 𝐼 → (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
10 df-vrgp 18105 . . . 4 varFGrp = (𝑖 ∈ V ↦ (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)))
11 vex 3198 . . . . 5 𝑖 ∈ V
1211mptex 6471 . . . 4 (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)) ∈ V
139, 10, 12fvmpt3i 6274 . . 3 (𝐼 ∈ V → (varFGrp𝐼) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
142, 13syl 17 . 2 (𝐼𝑉 → (varFGrp𝐼) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
151, 14syl5eq 2666 1 (𝐼𝑉𝑈 = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  Vcvv 3195  c0 3907  cop 4174  cmpt 4720  cfv 5876  [cec 7725  ⟨“cs1 13277   ~FG cefg 18100  varFGrpcvrgp 18102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ec 7729  df-vrgp 18105
This theorem is referenced by:  vrgpval  18161  vrgpf  18162
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