Theorem vrgpval 19679

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 19678	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))
4	3	fveq1d 6824	. 2 ⊢ (𝐼 ∈ 𝑉 → (𝑈‘𝐴) = ((𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )‘𝐴))
5		opeq1 4822	. . . . 5 ⊢ (𝑗 = 𝐴 → ⟨𝑗, ∅⟩ = ⟨𝐴, ∅⟩)
6	5	s1eqd 14509	. . . 4 ⊢ (𝑗 = 𝐴 → ⟨“⟨𝑗, ∅⟩”⟩ = ⟨“⟨𝐴, ∅⟩”⟩)
7	6	eceq1d 8662	. . 3 ⊢ (𝑗 = 𝐴 → [⟨“⟨𝑗, ∅⟩”⟩] ∼ = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
8		eqid 2731	. . 3 ⊢ (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ) = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )
9	1	fvexi 6836	. . . 4 ⊢ ∼ ∈ V
10		ecexg 8626	. . . 4 ⊢ ( ∼ ∈ V → [⟨“⟨𝐴, ∅⟩”⟩] ∼ ∈ V)
11	9, 10	ax-mp 5	. . 3 ⊢ [⟨“⟨𝐴, ∅⟩”⟩] ∼ ∈ V
12	7, 8, 11	fvmpt 6929	. 2 ⊢ (𝐴 ∈ 𝐼 → ((𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
13	4, 12	sylan9eq 2786	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ∅c0 4280  ⟨cop 4579   ↦ cmpt 5170  ‘cfv 6481  [cec 8620  ⟨“cs1 14503   ~_FG cefg 19618  var_FGrpcvrgp 19620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ec 8624  df-s1 14504  df-vrgp 19623

This theorem is referenced by:  vrgpinv  19681  frgpup2  19688  frgpup3lem  19689  frgpnabllem1  19785