Theorem xrneq12

Description: Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020)

	Ref	Expression
Assertion	xrneq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		xrneq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐶 ) )
2		xrneq2	⊢ ( 𝐶 = 𝐷 → ( 𝐵 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐷 ) )
3	1 2	sylan9eq	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐷 ) )