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Theorem vrgpval 18665
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 18664 . . 3 (𝐼𝑉𝑈 = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
43fveq1d 6498 . 2 (𝐼𝑉 → (𝑈𝐴) = ((𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] )‘𝐴))
5 opeq1 4673 . . . . 5 (𝑗 = 𝐴 → ⟨𝑗, ∅⟩ = ⟨𝐴, ∅⟩)
65s1eqd 13762 . . . 4 (𝑗 = 𝐴 → ⟨“⟨𝑗, ∅⟩”⟩ = ⟨“⟨𝐴, ∅⟩”⟩)
76eceq1d 8126 . . 3 (𝑗 = 𝐴 → [⟨“⟨𝑗, ∅⟩”⟩] = [⟨“⟨𝐴, ∅⟩”⟩] )
8 eqid 2771 . . 3 (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] )
91fvexi 6510 . . . 4 ∈ V
10 ecexg 8091 . . . 4 ( ∈ V → [⟨“⟨𝐴, ∅⟩”⟩] ∈ V)
119, 10ax-mp 5 . . 3 [⟨“⟨𝐴, ∅⟩”⟩] ∈ V
127, 8, 11fvmpt 6593 . 2 (𝐴𝐼 → ((𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] )‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
134, 12sylan9eq 2827 1 ((𝐼𝑉𝐴𝐼) → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1508  wcel 2051  Vcvv 3408  c0 4172  cop 4441  cmpt 5004  cfv 6185  [cec 8085  ⟨“cs1 13756   ~FG cefg 18602  varFGrpcvrgp 18604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ec 8089  df-s1 13757  df-vrgp 18607
This theorem is referenced by:  vrgpinv  18667  frgpup2  18674  frgpup3lem  18675  frgpnabllem1  18761
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