Theorem vrgpval 19981

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 19980	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))
4	3	fveq1d 6887	. 2 ⊢ (𝐼 ∈ 𝑉 → (𝑈‘𝐴) = ((𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )‘𝐴))
5		opeq1 4833	. . . . 5 ⊢ (𝑗 = 𝐴 → ⟨𝑗, ∅⟩ = ⟨𝐴, ∅⟩)
6	5	s1eqd 14748	. . . 4 ⊢ (𝑗 = 𝐴 → ⟨“⟨𝑗, ∅⟩”⟩ = ⟨“⟨𝐴, ∅⟩”⟩)
7	6	eceq1d 8758	. . 3 ⊢ (𝑗 = 𝐴 → [⟨“⟨𝑗, ∅⟩”⟩] ∼ = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
8		eqid 2761	. . 3 ⊢ (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ) = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )
9	1	fvexi 6899	. . . 4 ⊢ ∼ ∈ V
10		ecexg 8721	. . . 4 ⊢ ( ∼ ∈ V → [⟨“⟨𝐴, ∅⟩”⟩] ∼ ∈ V)
11	9, 10	ax-mp 5	. . 3 ⊢ [⟨“⟨𝐴, ∅⟩”⟩] ∼ ∈ V
12	7, 8, 11	fvmpt 6993	. 2 ⊢ (𝐴 ∈ 𝐼 → ((𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
13	4, 12	sylan9eq 2816	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ∅c0 4279  ⟨cop 4590   ↦ cmpt 5186  ‘cfv 6538  [cec 8715  ⟨“cs1 14742   ~_FG cefg 19920  var_FGrpcvrgp 19922

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ec 8719  df-s1 14743  df-vrgp 19925

This theorem is used by:  vrgpinv  19983  frgpup2  19990  frgpup3lem  19991  frgpnabllem1  20087