Theorem fvpr1g

Description: The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017)

	Ref	Expression
Assertion	fvpr1g	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵 ) → ( { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } ‘ 𝐴 ) = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		df-pr	⊢ { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } = ( { ⟨ 𝐴 , 𝐶 ⟩ } ∪ { ⟨ 𝐵 , 𝐷 ⟩ } )
2	1	fveq1i	⊢ ( { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } ‘ 𝐴 ) = ( ( { ⟨ 𝐴 , 𝐶 ⟩ } ∪ { ⟨ 𝐵 , 𝐷 ⟩ } ) ‘ 𝐴 )
3		necom	⊢ ( 𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴 )
4		fvunsn	⊢ ( 𝐵 ≠ 𝐴 → ( ( { ⟨ 𝐴 , 𝐶 ⟩ } ∪ { ⟨ 𝐵 , 𝐷 ⟩ } ) ‘ 𝐴 ) = ( { ⟨ 𝐴 , 𝐶 ⟩ } ‘ 𝐴 ) )
5	3 4	sylbi	⊢ ( 𝐴 ≠ 𝐵 → ( ( { ⟨ 𝐴 , 𝐶 ⟩ } ∪ { ⟨ 𝐵 , 𝐷 ⟩ } ) ‘ 𝐴 ) = ( { ⟨ 𝐴 , 𝐶 ⟩ } ‘ 𝐴 ) )
6	2 5	eqtrid	⊢ ( 𝐴 ≠ 𝐵 → ( { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } ‘ 𝐴 ) = ( { ⟨ 𝐴 , 𝐶 ⟩ } ‘ 𝐴 ) )
7	6	3ad2ant3	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵 ) → ( { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } ‘ 𝐴 ) = ( { ⟨ 𝐴 , 𝐶 ⟩ } ‘ 𝐴 ) )
8		fvsng	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( { ⟨ 𝐴 , 𝐶 ⟩ } ‘ 𝐴 ) = 𝐶 )
9	8	3adant3	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵 ) → ( { ⟨ 𝐴 , 𝐶 ⟩ } ‘ 𝐴 ) = 𝐶 )
10	7 9	eqtrd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵 ) → ( { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } ‘ 𝐴 ) = 𝐶 )