Theorem vrgpval 19704

Description: The value of the generating elements of a free group. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

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

Assertion

Ref	Expression
vrgpval	⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )

Proof of Theorem vrgpval

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vrgpfval.r	. . . 4 ⊢ ∼ = ( ~FG ‘𝐼)
2		vrgpfval.u	. . . 4 ⊢ 𝑈 = (varFGrp‘𝐼)
3	1, 2	vrgpfval 19703	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))
4	3	fveq1d 6863	. 2 ⊢ (𝐼 ∈ 𝑉 → (𝑈‘𝐴) = ((𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )‘𝐴))
5		opeq1 4840	. . . . 5 ⊢ (𝑗 = 𝐴 → ⟨𝑗, ∅⟩ = ⟨𝐴, ∅⟩)
6	5	s1eqd 14573	. . . 4 ⊢ (𝑗 = 𝐴 → ⟨“⟨𝑗, ∅⟩”⟩ = ⟨“⟨𝐴, ∅⟩”⟩)
7	6	eceq1d 8714	. . 3 ⊢ (𝑗 = 𝐴 → [⟨“⟨𝑗, ∅⟩”⟩] ∼ = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
8		eqid 2730	. . 3 ⊢ (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ) = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )
9	1	fvexi 6875	. . . 4 ⊢ ∼ ∈ V
10		ecexg 8678	. . . 4 ⊢ ( ∼ ∈ V → [⟨“⟨𝐴, ∅⟩”⟩] ∼ ∈ V)
11	9, 10	ax-mp 5	. . 3 ⊢ [⟨“⟨𝐴, ∅⟩”⟩] ∼ ∈ V
12	7, 8, 11	fvmpt 6971	. 2 ⊢ (𝐴 ∈ 𝐼 → ((𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
13	4, 12	sylan9eq 2785	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ∅c0 4299  ⟨cop 4598   ↦ cmpt 5191  ‘cfv 6514  [cec 8672  ⟨“cs1 14567   ~_FG cefg 19643  var_FGrpcvrgp 19645

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ec 8676  df-s1 14568  df-vrgp 19648

This theorem is referenced by:  vrgpinv  19706  frgpup2  19713  frgpup3lem  19714  frgpnabllem1  19810