Theorem vrgpval 19694

Description: The value of the generating elements of a free group. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
vrgpfval.r	⊢ ∼ = ( ~FG ‘𝐼)
vrgpfval.u	⊢ 𝑈 = (varFGrp‘𝐼)

Assertion

Ref	Expression
vrgpval	⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )

Proof of Theorem vrgpval

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vrgpfval.r	. . . 4 ⊢ ∼ = ( ~FG ‘𝐼)
2		vrgpfval.u	. . . 4 ⊢ 𝑈 = (varFGrp‘𝐼)
3	1, 2	vrgpfval 19693	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))
4	3	fveq1d 6834	. 2 ⊢ (𝐼 ∈ 𝑉 → (𝑈‘𝐴) = ((𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )‘𝐴))
5		opeq1 4827	. . . . 5 ⊢ (𝑗 = 𝐴 → ⟨𝑗, ∅⟩ = ⟨𝐴, ∅⟩)
6	5	s1eqd 14523	. . . 4 ⊢ (𝑗 = 𝐴 → ⟨“⟨𝑗, ∅⟩”⟩ = ⟨“⟨𝐴, ∅⟩”⟩)
7	6	eceq1d 8673	. . 3 ⊢ (𝑗 = 𝐴 → [⟨“⟨𝑗, ∅⟩”⟩] ∼ = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
8		eqid 2734	. . 3 ⊢ (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ) = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )
9	1	fvexi 6846	. . . 4 ⊢ ∼ ∈ V
10		ecexg 8637	. . . 4 ⊢ ( ∼ ∈ V → [⟨“⟨𝐴, ∅⟩”⟩] ∼ ∈ V)
11	9, 10	ax-mp 5	. . 3 ⊢ [⟨“⟨𝐴, ∅⟩”⟩] ∼ ∈ V
12	7, 8, 11	fvmpt 6939	. 2 ⊢ (𝐴 ∈ 𝐼 → ((𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
13	4, 12	sylan9eq 2789	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ∅c0 4283  ⟨cop 4584   ↦ cmpt 5177  ‘cfv 6490  [cec 8631  ⟨“cs1 14517   ~_FG cefg 19633  var_FGrpcvrgp 19635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ec 8635  df-s1 14518  df-vrgp 19638

This theorem is referenced by:  vrgpinv  19696  frgpup2  19703  frgpup3lem  19704  frgpnabllem1  19800