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Theorem vrgpfval 18386
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 3364 . . 3 (𝐼𝑉𝐼 ∈ V)
3 id 22 . . . . 5 (𝑖 = 𝐼𝑖 = 𝐼)
4 fveq2 6333 . . . . . . 7 (𝑖 = 𝐼 → ( ~FG𝑖) = ( ~FG𝐼))
5 vrgpfval.r . . . . . . 7 = ( ~FG𝐼)
64, 5syl6eqr 2823 . . . . . 6 (𝑖 = 𝐼 → ( ~FG𝑖) = )
7 eceq2 7940 . . . . . 6 (( ~FG𝑖) = → [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖) = [⟨“⟨𝑗, ∅⟩”⟩] )
86, 7syl 17 . . . . 5 (𝑖 = 𝐼 → [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖) = [⟨“⟨𝑗, ∅⟩”⟩] )
93, 8mpteq12dv 4868 . . . 4 (𝑖 = 𝐼 → (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
10 df-vrgp 18331 . . . 4 varFGrp = (𝑖 ∈ V ↦ (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)))
11 vex 3354 . . . . 5 𝑖 ∈ V
1211mptex 6633 . . . 4 (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)) ∈ V
139, 10, 12fvmpt3i 6431 . . 3 (𝐼 ∈ V → (varFGrp𝐼) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
142, 13syl 17 . 2 (𝐼𝑉 → (varFGrp𝐼) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
151, 14syl5eq 2817 1 (𝐼𝑉𝑈 = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351  c0 4063  cop 4323  cmpt 4864  cfv 6030  [cec 7898  ⟨“cs1 13490   ~FG cefg 18326  varFGrpcvrgp 18328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ec 7902  df-vrgp 18331
This theorem is referenced by:  vrgpval  18387  vrgpf  18388
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