Theorem fvtp2

Description: The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011)

	Ref	Expression
Hypotheses	fvtp2.1	⊢ 𝐵 ∈ V
	fvtp2.4	⊢ 𝐸 ∈ V
Assertion	fvtp2	⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) → ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐵 ) = 𝐸 )

Proof

Step	Hyp	Ref	Expression
1		fvtp2.1	⊢ 𝐵 ∈ V
2		fvtp2.4	⊢ 𝐸 ∈ V
3		tprot	⊢ { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } = { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ }
4	3	fveq1i	⊢ ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐵 ) = ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐵 )
5		necom	⊢ ( 𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴 )
6	1 2	fvtp1	⊢ ( ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴 ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐸 )
7	6	ancoms	⊢ ( ( 𝐵 ≠ 𝐴 ∧ 𝐵 ≠ 𝐶 ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐸 )
8	5 7	sylanb	⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐸 )
9	4 8	eqtrid	⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) → ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐵 ) = 𝐸 )