Theorem orbstaval

Description: Value of the function at a given equivalence class element. (Contributed by Mario Carneiro, 15-Jan-2015) (Proof shortened by Mario Carneiro, 23-Dec-2016)

	Ref	Expression
Hypotheses	gasta.1	⊢ 𝑋 = ( Base ‘ 𝐺 )
	gasta.2	⊢ 𝐻 = { 𝑢 ∈ 𝑋 ∣ ( 𝑢 ⊕ 𝐴 ) = 𝐴 }
	orbsta.r	⊢ ∼ = ( 𝐺 ~QG 𝐻 )
	orbsta.f	⊢ 𝐹 = ran ( 𝑘 ∈ 𝑋 ↦ ⟨ [ 𝑘 ] ∼ , ( 𝑘 ⊕ 𝐴 ) ⟩ )
Assertion	orbstaval	⊢ ( ( ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) ∧ 𝐴 ∈ 𝑌 ) ∧ 𝐵 ∈ 𝑋 ) → ( 𝐹 ‘ [ 𝐵 ] ∼ ) = ( 𝐵 ⊕ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		gasta.1	⊢ 𝑋 = ( Base ‘ 𝐺 )
2		gasta.2	⊢ 𝐻 = { 𝑢 ∈ 𝑋 ∣ ( 𝑢 ⊕ 𝐴 ) = 𝐴 }
3		orbsta.r	⊢ ∼ = ( 𝐺 ~QG 𝐻 )
4		orbsta.f	⊢ 𝐹 = ran ( 𝑘 ∈ 𝑋 ↦ ⟨ [ 𝑘 ] ∼ , ( 𝑘 ⊕ 𝐴 ) ⟩ )
5		ovexd	⊢ ( ( ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) ∧ 𝐴 ∈ 𝑌 ) ∧ 𝑘 ∈ 𝑋 ) → ( 𝑘 ⊕ 𝐴 ) ∈ V )
6	1 2	gastacl	⊢ ( ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) ∧ 𝐴 ∈ 𝑌 ) → 𝐻 ∈ ( SubGrp ‘ 𝐺 ) )
7	1 3	eqger	⊢ ( 𝐻 ∈ ( SubGrp ‘ 𝐺 ) → ∼ Er 𝑋 )
8	6 7	syl	⊢ ( ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) ∧ 𝐴 ∈ 𝑌 ) → ∼ Er 𝑋 )
9	1	fvexi	⊢ 𝑋 ∈ V
10	9	a1i	⊢ ( ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) ∧ 𝐴 ∈ 𝑌 ) → 𝑋 ∈ V )
11		oveq1	⊢ ( 𝑘 = 𝐵 → ( 𝑘 ⊕ 𝐴 ) = ( 𝐵 ⊕ 𝐴 ) )
12	1 2 3 4	orbstafun	⊢ ( ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) ∧ 𝐴 ∈ 𝑌 ) → Fun 𝐹 )
13	4 5 8 10 11 12	qliftval	⊢ ( ( ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) ∧ 𝐴 ∈ 𝑌 ) ∧ 𝐵 ∈ 𝑋 ) → ( 𝐹 ‘ [ 𝐵 ] ∼ ) = ( 𝐵 ⊕ 𝐴 ) )