Theorem xpid11

Description: The Cartesian square is a one-to-one construction. (Contributed by NM, 5-Nov-2006) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	xpid11	⊢ ( ( 𝐴 × 𝐴 ) = ( 𝐵 × 𝐵 ) ↔ 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		dmeq	⊢ ( ( 𝐴 × 𝐴 ) = ( 𝐵 × 𝐵 ) → dom ( 𝐴 × 𝐴 ) = dom ( 𝐵 × 𝐵 ) )
2		dmxpid	⊢ dom ( 𝐴 × 𝐴 ) = 𝐴
3		dmxpid	⊢ dom ( 𝐵 × 𝐵 ) = 𝐵
4	1 2 3	3eqtr3g	⊢ ( ( 𝐴 × 𝐴 ) = ( 𝐵 × 𝐵 ) → 𝐴 = 𝐵 )
5		xpeq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐵 ) → ( 𝐴 × 𝐴 ) = ( 𝐵 × 𝐵 ) )
6	5	anidms	⊢ ( 𝐴 = 𝐵 → ( 𝐴 × 𝐴 ) = ( 𝐵 × 𝐵 ) )
7	4 6	impbii	⊢ ( ( 𝐴 × 𝐴 ) = ( 𝐵 × 𝐵 ) ↔ 𝐴 = 𝐵 )