Theorem reseq1

Description: Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994)

	Ref	Expression
Assertion	reseq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ↾ 𝐶 ) = ( 𝐵 ↾ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ineq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∩ ( 𝐶 × V ) ) = ( 𝐵 ∩ ( 𝐶 × V ) ) )
2		df-res	⊢ ( 𝐴 ↾ 𝐶 ) = ( 𝐴 ∩ ( 𝐶 × V ) )
3		df-res	⊢ ( 𝐵 ↾ 𝐶 ) = ( 𝐵 ∩ ( 𝐶 × V ) )
4	1 2 3	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ↾ 𝐶 ) = ( 𝐵 ↾ 𝐶 ) )