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Theorem vrgpval 19786
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 19785 . . 3 (𝐼𝑉𝑈 = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
43fveq1d 6907 . 2 (𝐼𝑉 → (𝑈𝐴) = ((𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] )‘𝐴))
5 opeq1 4872 . . . . 5 (𝑗 = 𝐴 → ⟨𝑗, ∅⟩ = ⟨𝐴, ∅⟩)
65s1eqd 14640 . . . 4 (𝑗 = 𝐴 → ⟨“⟨𝑗, ∅⟩”⟩ = ⟨“⟨𝐴, ∅⟩”⟩)
76eceq1d 8786 . . 3 (𝑗 = 𝐴 → [⟨“⟨𝑗, ∅⟩”⟩] = [⟨“⟨𝐴, ∅⟩”⟩] )
8 eqid 2736 . . 3 (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] )
91fvexi 6919 . . . 4 ∈ V
10 ecexg 8750 . . . 4 ( ∈ V → [⟨“⟨𝐴, ∅⟩”⟩] ∈ V)
119, 10ax-mp 5 . . 3 [⟨“⟨𝐴, ∅⟩”⟩] ∈ V
127, 8, 11fvmpt 7015 . 2 (𝐴𝐼 → ((𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] )‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
134, 12sylan9eq 2796 1 ((𝐼𝑉𝐴𝐼) → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  c0 4332  cop 4631  cmpt 5224  cfv 6560  [cec 8744  ⟨“cs1 14634   ~FG cefg 19725  varFGrpcvrgp 19727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ec 8748  df-s1 14635  df-vrgp 19730
This theorem is referenced by:  vrgpinv  19788  frgpup2  19795  frgpup3lem  19796  frgpnabllem1  19892
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