Theorem brdomaing

Description: Closed form of brdomain . (Contributed by Scott Fenton, 2-May-2014)

	Ref	Expression
Assertion	brdomaing	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 Domain 𝐵 ↔ 𝐵 = dom 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		breq1	⊢ ( 𝑎 = 𝐴 → ( 𝑎 Domain 𝑏 ↔ 𝐴 Domain 𝑏 ) )
2		dmeq	⊢ ( 𝑎 = 𝐴 → dom 𝑎 = dom 𝐴 )
3	2	eqeq2d	⊢ ( 𝑎 = 𝐴 → ( 𝑏 = dom 𝑎 ↔ 𝑏 = dom 𝐴 ) )
4	1 3	bibi12d	⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 Domain 𝑏 ↔ 𝑏 = dom 𝑎 ) ↔ ( 𝐴 Domain 𝑏 ↔ 𝑏 = dom 𝐴 ) ) )
5		breq2	⊢ ( 𝑏 = 𝐵 → ( 𝐴 Domain 𝑏 ↔ 𝐴 Domain 𝐵 ) )
6		eqeq1	⊢ ( 𝑏 = 𝐵 → ( 𝑏 = dom 𝐴 ↔ 𝐵 = dom 𝐴 ) )
7	5 6	bibi12d	⊢ ( 𝑏 = 𝐵 → ( ( 𝐴 Domain 𝑏 ↔ 𝑏 = dom 𝐴 ) ↔ ( 𝐴 Domain 𝐵 ↔ 𝐵 = dom 𝐴 ) ) )
8		vex	⊢ 𝑎 ∈ V
9		vex	⊢ 𝑏 ∈ V
10	8 9	brdomain	⊢ ( 𝑎 Domain 𝑏 ↔ 𝑏 = dom 𝑎 )
11	4 7 10	vtocl2g	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 Domain 𝐵 ↔ 𝐵 = dom 𝐴 ) )