Theorem vrgpfval 19664

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 3459	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		id 22	. . . . 5 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼)
4		fveq2 6826	. . . . . . 7 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼))
5		vrgpfval.r	. . . . . . 7 ⊢ ∼ = ( ~FG ‘𝐼)
6	4, 5	eqtr4di 2782	. . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ )
7	6	eceq2d 8675	. . . . 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 19609	. . . 4 ⊢ varFGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)))
10		vex 3442	. . . . 5 ⊢ 𝑖 ∈ V
11	10	mptex 7163	. . . 4 ⊢ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)) ∈ V
12	8, 9, 11	fvmpt3i 6939	. . 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 2776	1 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3438  ∅c0 4286  ⟨cop 4585   ↦ cmpt 5176  ‘cfv 6486  [cec 8630  ⟨“cs1 14521   ~_FG cefg 19604  var_FGrpcvrgp 19606

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ec 8634  df-vrgp 19609

This theorem is referenced by:  vrgpval  19665  vrgpf  19666