Theorem idresssidinxp

Description: Condition for the identity restriction to be a subclass of identity intersection with a Cartesian product. (Contributed by Peter Mazsa, 19-Jul-2018)

	Ref	Expression
Assertion	idresssidinxp	⊢ ( 𝐴 ⊆ 𝐵 → ( I ↾ 𝐴 ) ⊆ ( I ∩ ( 𝐴 × 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		resss	⊢ ( I ↾ 𝐴 ) ⊆ I
2	1	a1i	⊢ ( 𝐴 ⊆ 𝐵 → ( I ↾ 𝐴 ) ⊆ I )
3		idssxp	⊢ ( I ↾ 𝐴 ) ⊆ ( 𝐴 × 𝐴 )
4		xpss2	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 × 𝐴 ) ⊆ ( 𝐴 × 𝐵 ) )
5	3 4	sstrid	⊢ ( 𝐴 ⊆ 𝐵 → ( I ↾ 𝐴 ) ⊆ ( 𝐴 × 𝐵 ) )
6	2 5	ssind	⊢ ( 𝐴 ⊆ 𝐵 → ( I ↾ 𝐴 ) ⊆ ( I ∩ ( 𝐴 × 𝐵 ) ) )