Theorem xrneq1

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

	Ref	Expression
Assertion	xrneq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		coeq2	⊢ ( 𝐴 = 𝐵 → ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝐴 ) = ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝐵 ) )
2	1	ineq1d	⊢ ( 𝐴 = 𝐵 → ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝐴 ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝐶 ) ) = ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝐵 ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝐶 ) ) )
3		df-xrn	⊢ ( 𝐴 ⋉ 𝐶 ) = ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝐴 ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝐶 ) )
4		df-xrn	⊢ ( 𝐵 ⋉ 𝐶 ) = ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝐵 ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝐶 ) )
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐶 ) )