Theorem vrgpfval 19784

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 3501	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		id 22	. . . . 5 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼)
4		fveq2 6906	. . . . . . 7 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼))
5		vrgpfval.r	. . . . . . 7 ⊢ ∼ = ( ~FG ‘𝐼)
6	4, 5	eqtr4di 2795	. . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ )
7	6	eceq2d 8788	. . . . 5 ⊢ (𝑖 = 𝐼 → [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖) = [⟨“⟨𝑗, ∅⟩”⟩] ∼ )
8	3, 7	mpteq12dv 5233	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)) = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))
9		df-vrgp 19729	. . . 4 ⊢ varFGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)))
10		vex 3484	. . . . 5 ⊢ 𝑖 ∈ V
11	10	mptex 7243	. . . 4 ⊢ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)) ∈ V
12	8, 9, 11	fvmpt3i 7021	. . 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 2789	1 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ∅c0 4333  ⟨cop 4632   ↦ cmpt 5225  ‘cfv 6561  [cec 8743  ⟨“cs1 14633   ~_FG cefg 19724  var_FGrpcvrgp 19726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ec 8747  df-vrgp 19729

This theorem is referenced by:  vrgpval  19785  vrgpf  19786