Theorem reseq2

Description: Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994)

	Ref	Expression
Assertion	reseq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ↾ 𝐴 ) = ( 𝐶 ↾ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		xpeq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 × V ) = ( 𝐵 × V ) )
2	1	ineq2d	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ∩ ( 𝐴 × V ) ) = ( 𝐶 ∩ ( 𝐵 × V ) ) )
3		df-res	⊢ ( 𝐶 ↾ 𝐴 ) = ( 𝐶 ∩ ( 𝐴 × V ) )
4		df-res	⊢ ( 𝐶 ↾ 𝐵 ) = ( 𝐶 ∩ ( 𝐵 × V ) )
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ↾ 𝐴 ) = ( 𝐶 ↾ 𝐵 ) )