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Theorem vrgpval 18387
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 18386 . . 3 (𝐼𝑉𝑈 = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))
43fveq1d 6335 . 2 (𝐼𝑉 → (𝑈𝐴) = ((𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] )‘𝐴))
5 opeq1 4540 . . . . 5 (𝑗 = 𝐴 → ⟨𝑗, ∅⟩ = ⟨𝐴, ∅⟩)
65s1eqd 13581 . . . 4 (𝑗 = 𝐴 → ⟨“⟨𝑗, ∅⟩”⟩ = ⟨“⟨𝐴, ∅⟩”⟩)
76eceq1d 7939 . . 3 (𝑗 = 𝐴 → [⟨“⟨𝑗, ∅⟩”⟩] = [⟨“⟨𝐴, ∅⟩”⟩] )
8 eqid 2771 . . 3 (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ) = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] )
91fvexi 6345 . . . 4 ∈ V
10 ecexg 7904 . . . 4 ( ∈ V → [⟨“⟨𝐴, ∅⟩”⟩] ∈ V)
119, 10ax-mp 5 . . 3 [⟨“⟨𝐴, ∅⟩”⟩] ∈ V
127, 8, 11fvmpt 6426 . 2 (𝐴𝐼 → ((𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] )‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
134, 12sylan9eq 2825 1 ((𝐼𝑉𝐴𝐼) → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = 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-8 2147  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  ax-un 7100
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-s1 13498  df-vrgp 18331
This theorem is referenced by:  vrgpinv  18389  frgpup2  18396  frgpup3lem  18397  frgpnabllem1  18483
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