Theorem dmxpid

Description: The domain of a Cartesian square. (Contributed by NM, 28-Jul-1995)

	Ref	Expression
Assertion	dmxpid	⊢ dom ( 𝐴 × 𝐴 ) = 𝐴

Proof

Step	Hyp	Ref	Expression
1		dm0	⊢ dom ∅ = ∅
2		xpeq1	⊢ ( 𝐴 = ∅ → ( 𝐴 × 𝐴 ) = ( ∅ × 𝐴 ) )
3		0xp	⊢ ( ∅ × 𝐴 ) = ∅
4	2 3	eqtrdi	⊢ ( 𝐴 = ∅ → ( 𝐴 × 𝐴 ) = ∅ )
5	4	dmeqd	⊢ ( 𝐴 = ∅ → dom ( 𝐴 × 𝐴 ) = dom ∅ )
6		id	⊢ ( 𝐴 = ∅ → 𝐴 = ∅ )
7	1 5 6	3eqtr4a	⊢ ( 𝐴 = ∅ → dom ( 𝐴 × 𝐴 ) = 𝐴 )
8		dmxp	⊢ ( 𝐴 ≠ ∅ → dom ( 𝐴 × 𝐴 ) = 𝐴 )
9	7 8	pm2.61ine	⊢ dom ( 𝐴 × 𝐴 ) = 𝐴