Theorem ex-fv

Description: Example for df-fv . Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015)

	Ref	Expression
Assertion	ex-fv	⊢ ( 𝐹 = { ⟨ 2 , 6 ⟩ , ⟨ 3 , 9 ⟩ } → ( 𝐹 ‘ 3 ) = 9 )

Proof

Step	Hyp	Ref	Expression
1		fveq1	⊢ ( 𝐹 = { ⟨ 2 , 6 ⟩ , ⟨ 3 , 9 ⟩ } → ( 𝐹 ‘ 3 ) = ( { ⟨ 2 , 6 ⟩ , ⟨ 3 , 9 ⟩ } ‘ 3 ) )
2		2re	⊢ 2 ∈ ℝ
3		2lt3	⊢ 2 < 3
4	2 3	ltneii	⊢ 2 ≠ 3
5		3ex	⊢ 3 ∈ V
6		9re	⊢ 9 ∈ ℝ
7	6	elexi	⊢ 9 ∈ V
8	5 7	fvpr2	⊢ ( 2 ≠ 3 → ( { ⟨ 2 , 6 ⟩ , ⟨ 3 , 9 ⟩ } ‘ 3 ) = 9 )
9	4 8	ax-mp	⊢ ( { ⟨ 2 , 6 ⟩ , ⟨ 3 , 9 ⟩ } ‘ 3 ) = 9
10	1 9	eqtrdi	⊢ ( 𝐹 = { ⟨ 2 , 6 ⟩ , ⟨ 3 , 9 ⟩ } → ( 𝐹 ‘ 3 ) = 9 )