Theorem vrgpfval 19741

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 3450	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		id 22	. . . . 5 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼)
4		fveq2 6840	. . . . . . 7 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼))
5		vrgpfval.r	. . . . . . 7 ⊢ ∼ = ( ~FG ‘𝐼)
6	4, 5	eqtr4di 2789	. . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ )
7	6	eceq2d 8687	. . . . 5 ⊢ (𝑖 = 𝐼 → [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖) = [⟨“⟨𝑗, ∅⟩”⟩] ∼ )
8	3, 7	mpteq12dv 5172	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)) = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))
9		df-vrgp 19686	. . . 4 ⊢ varFGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)))
10		vex 3433	. . . . 5 ⊢ 𝑖 ∈ V
11	10	mptex 7178	. . . 4 ⊢ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)) ∈ V
12	8, 9, 11	fvmpt3i 6953	. . 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 2783	1 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ∅c0 4273  ⟨cop 4573   ↦ cmpt 5166  ‘cfv 6498  [cec 8641  ⟨“cs1 14558   ~_FG cefg 19681  var_FGrpcvrgp 19683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ec 8645  df-vrgp 19686

This theorem is referenced by:  vrgpval  19742  vrgpf  19743