Theorem rescom

Description: Commutative law for restriction. (Contributed by NM, 27-Mar-1998)

	Ref	Expression
Assertion	rescom	⊢ ( ( 𝐴 ↾ 𝐵 ) ↾ 𝐶 ) = ( ( 𝐴 ↾ 𝐶 ) ↾ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		incom	⊢ ( 𝐵 ∩ 𝐶 ) = ( 𝐶 ∩ 𝐵 )
2	1	reseq2i	⊢ ( 𝐴 ↾ ( 𝐵 ∩ 𝐶 ) ) = ( 𝐴 ↾ ( 𝐶 ∩ 𝐵 ) )
3		resres	⊢ ( ( 𝐴 ↾ 𝐵 ) ↾ 𝐶 ) = ( 𝐴 ↾ ( 𝐵 ∩ 𝐶 ) )
4		resres	⊢ ( ( 𝐴 ↾ 𝐶 ) ↾ 𝐵 ) = ( 𝐴 ↾ ( 𝐶 ∩ 𝐵 ) )
5	2 3 4	3eqtr4i	⊢ ( ( 𝐴 ↾ 𝐵 ) ↾ 𝐶 ) = ( ( 𝐴 ↾ 𝐶 ) ↾ 𝐵 )