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Theorem vrgpval 19800
Description: The value of the generating elements of a free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
vrgpfval.r = ( ~FG𝐼)
vrgpfval.u 𝑈 = (varFGrp𝐼)
Assertion
Ref Expression
vrgpval ((𝐼𝑉𝐴𝐼) → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )

Proof of Theorem vrgpval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 vrgpfval.r . . . 4 = ( ~FG𝐼)
2 vrgpfval.u . . . 4 𝑈 = (varFGrp𝐼)
31, 2vrgpfval 19799 . . 3 (𝐼𝑉𝑈 = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
43fveq1d 6909 . 2 (𝐼𝑉 → (𝑈𝐴) = ((𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] )‘𝐴))
5 opeq1 4878 . . . . 5 (𝑗 = 𝐴 → ⟨𝑗, ∅⟩ = ⟨𝐴, ∅⟩)
65s1eqd 14636 . . . 4 (𝑗 = 𝐴 → ⟨“⟨𝑗, ∅⟩”⟩ = ⟨“⟨𝐴, ∅⟩”⟩)
76eceq1d 8784 . . 3 (𝑗 = 𝐴 → [⟨“⟨𝑗, ∅⟩”⟩] = [⟨“⟨𝐴, ∅⟩”⟩] )
8 eqid 2735 . . 3 (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] )
91fvexi 6921 . . . 4 ∈ V
10 ecexg 8748 . . . 4 ( ∈ V → [⟨“⟨𝐴, ∅⟩”⟩] ∈ V)
119, 10ax-mp 5 . . 3 [⟨“⟨𝐴, ∅⟩”⟩] ∈ V
127, 8, 11fvmpt 7016 . 2 (𝐴𝐼 → ((𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] )‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
134, 12sylan9eq 2795 1 ((𝐼𝑉𝐴𝐼) → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = 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  ax-un 7754
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-s1 14631  df-vrgp 19744
This theorem is referenced by:  vrgpinv  19802  frgpup2  19809  frgpup3lem  19810  frgpnabllem1  19906
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