Theorem vrgpfval 19686

Description: The canonical injection from the generating set 𝐼 to the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

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

Assertion

Ref	Expression
vrgpfval	⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))

Distinct variable groups:   𝑗,𝐼   ∼ ,𝑗   𝑗,𝑉

Allowed substitution hint:   𝑈(𝑗)

Proof of Theorem vrgpfval

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vrgpfval.u	. 2 ⊢ 𝑈 = (varFGrp‘𝐼)
2		elex 3458	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		id 22	. . . . 5 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼)
4		fveq2 6831	. . . . . . 7 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼))
5		vrgpfval.r	. . . . . . 7 ⊢ ∼ = ( ~FG ‘𝐼)
6	4, 5	eqtr4di 2786	. . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ )
7	6	eceq2d 8674	. . . . 5 ⊢ (𝑖 = 𝐼 → [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖) = [⟨“⟨𝑗, ∅⟩”⟩] ∼ )
8	3, 7	mpteq12dv 5182	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)) = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))
9		df-vrgp 19631	. . . 4 ⊢ varFGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)))
10		vex 3441	. . . . 5 ⊢ 𝑖 ∈ V
11	10	mptex 7166	. . . 4 ⊢ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)) ∈ V
12	8, 9, 11	fvmpt3i 6943	. . 3 ⊢ (𝐼 ∈ V → (varFGrp‘𝐼) = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))
13	2, 12	syl 17	. 2 ⊢ (𝐼 ∈ 𝑉 → (varFGrp‘𝐼) = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))
14	1, 13	eqtrid 2780	1 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ∅c0 4282  ⟨cop 4583   ↦ cmpt 5176  ‘cfv 6489  [cec 8629  ⟨“cs1 14510   ~_FG cefg 19626  var_FGrpcvrgp 19628

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 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ec 8633  df-vrgp 19631

This theorem is referenced by:  vrgpval  19687  vrgpf  19688