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Theorem vrgpfval 19686
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 3487 . . 3 (𝐼𝑉𝐼 ∈ V)
3 id 22 . . . . 5 (𝑖 = 𝐼𝑖 = 𝐼)
4 fveq2 6885 . . . . . . 7 (𝑖 = 𝐼 → ( ~FG𝑖) = ( ~FG𝐼))
5 vrgpfval.r . . . . . . 7 = ( ~FG𝐼)
64, 5eqtr4di 2784 . . . . . 6 (𝑖 = 𝐼 → ( ~FG𝑖) = )
76eceq2d 8747 . . . . 5 (𝑖 = 𝐼 → [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖) = [⟨“⟨𝑗, ∅⟩”⟩] )
83, 7mpteq12dv 5232 . . . 4 (𝑖 = 𝐼 → (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
9 df-vrgp 19631 . . . 4 varFGrp = (𝑖 ∈ V ↦ (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)))
10 vex 3472 . . . . 5 𝑖 ∈ V
1110mptex 7220 . . . 4 (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)) ∈ V
128, 9, 11fvmpt3i 6997 . . 3 (𝐼 ∈ V → (varFGrp𝐼) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
132, 12syl 17 . 2 (𝐼𝑉 → (varFGrp𝐼) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
141, 13eqtrid 2778 1 (𝐼𝑉𝑈 = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3468  c0 4317  cop 4629  cmpt 5224  cfv 6537  [cec 8703  ⟨“cs1 14551   ~FG cefg 19626  varFGrpcvrgp 19628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ec 8707  df-vrgp 19631
This theorem is referenced by:  vrgpval  19687  vrgpf  19688
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