Theorem fvtp2g

Description: The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017)

	Ref	Expression
Assertion	fvtp2g	⊢ ( ( ( 𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ) → ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐵 ) = 𝐸 )

Proof

Step	Hyp	Ref	Expression
1		tprot	⊢ { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } = { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ }
2	1	fveq1i	⊢ ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐵 ) = ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐵 )
3		necom	⊢ ( 𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴 )
4		fvtp1g	⊢ ( ( ( 𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴 ) ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐸 )
5	4	expcom	⊢ ( ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴 ) → ( ( 𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐸 ) )
6	5	ancoms	⊢ ( ( 𝐵 ≠ 𝐴 ∧ 𝐵 ≠ 𝐶 ) → ( ( 𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐸 ) )
7	3 6	sylanb	⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) → ( ( 𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐸 ) )
8	7	impcom	⊢ ( ( ( 𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐸 )
9	2 8	eqtrid	⊢ ( ( ( 𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ) → ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐵 ) = 𝐸 )