Theorem vrgpfval 19736

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 3454	. . 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 2794	. . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ )
7	6	eceq2d 8681	. . . . 5 ⊢ (𝑖 = 𝐼 → [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖) = [⟨“⟨𝑗, ∅⟩”⟩] ∼ )
8	3, 7	mpteq12dv 5162	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)) = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))
9		df-vrgp 19681	. . . 4 ⊢ varFGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)))
10		vex 3437	. . . . 5 ⊢ 𝑖 ∈ V
11	10	mptex 7171	. . . 4 ⊢ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)) ∈ V
12	8, 9, 11	fvmpt3i 6945	. . 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 2788	1 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121  Vcvv 3433  ∅c0 4264  ⟨cop 4564   ↦ cmpt 5156  ‘cfv 6489  [cec 8635  ⟨“cs1 14553   ~_FG cefg 19676  var_FGrpcvrgp 19678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  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 8639  df-vrgp 19681

This theorem is referenced by:  vrgpval  19737  vrgpf  19738