Theorem brimageg

Description: Closed form of brimage . (Contributed by Scott Fenton, 4-Apr-2014) (Revised by Mario Carneiro, 19-Apr-2014)

	Ref	Expression
Assertion	brimageg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 Image 𝑅 𝐵 ↔ 𝐵 = ( 𝑅 “ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		breq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 Image 𝑅 𝑦 ↔ 𝐴 Image 𝑅 𝑦 ) )
2		imaeq2	⊢ ( 𝑥 = 𝐴 → ( 𝑅 “ 𝑥 ) = ( 𝑅 “ 𝐴 ) )
3	2	eqeq2d	⊢ ( 𝑥 = 𝐴 → ( 𝑦 = ( 𝑅 “ 𝑥 ) ↔ 𝑦 = ( 𝑅 “ 𝐴 ) ) )
4	1 3	bibi12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 Image 𝑅 𝑦 ↔ 𝑦 = ( 𝑅 “ 𝑥 ) ) ↔ ( 𝐴 Image 𝑅 𝑦 ↔ 𝑦 = ( 𝑅 “ 𝐴 ) ) ) )
5		breq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 Image 𝑅 𝑦 ↔ 𝐴 Image 𝑅 𝐵 ) )
6		eqeq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 = ( 𝑅 “ 𝐴 ) ↔ 𝐵 = ( 𝑅 “ 𝐴 ) ) )
7	5 6	bibi12d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 Image 𝑅 𝑦 ↔ 𝑦 = ( 𝑅 “ 𝐴 ) ) ↔ ( 𝐴 Image 𝑅 𝐵 ↔ 𝐵 = ( 𝑅 “ 𝐴 ) ) ) )
8		vex	⊢ 𝑥 ∈ V
9		vex	⊢ 𝑦 ∈ V
10	8 9	brimage	⊢ ( 𝑥 Image 𝑅 𝑦 ↔ 𝑦 = ( 𝑅 “ 𝑥 ) )
11	4 7 10	vtocl2g	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 Image 𝑅 𝐵 ↔ 𝐵 = ( 𝑅 “ 𝐴 ) ) )