Theorem ressabs

Description: Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015)

	Ref	Expression
Assertion	ressabs	$$⊢\left(A\in X\wedge B\subseteq A\right)\to \left(W{\upharpoonright }_{𝑠}A\right){\upharpoonright }_{𝑠}B=W{\upharpoonright }_{𝑠}B$$

Proof

Step	Hyp	Ref	Expression
1		ssexg	$$⊢\left(B\subseteq A\wedge A\in X\right)\to B\in \mathrm{V}$$
2	1	ancoms	$$⊢\left(A\in X\wedge B\subseteq A\right)\to B\in \mathrm{V}$$
3		ressress	$$⊢\left(A\in X\wedge B\in \mathrm{V}\right)\to \left(W{\upharpoonright }_{𝑠}A\right){\upharpoonright }_{𝑠}B=W{\upharpoonright }_{𝑠}\left(A\cap B\right)$$
4	2 3	syldan	$$⊢\left(A\in X\wedge B\subseteq A\right)\to \left(W{\upharpoonright }_{𝑠}A\right){\upharpoonright }_{𝑠}B=W{\upharpoonright }_{𝑠}\left(A\cap B\right)$$
5		simpr	$$⊢\left(A\in X\wedge B\subseteq A\right)\to B\subseteq A$$
6		sseqin2	$$⊢B\subseteq A\leftrightarrow A\cap B=B$$
7	5 6	sylib	$$⊢\left(A\in X\wedge B\subseteq A\right)\to A\cap B=B$$
8	7	oveq2d	$$⊢\left(A\in X\wedge B\subseteq A\right)\to W{\upharpoonright }_{𝑠}\left(A\cap B\right)=W{\upharpoonright }_{𝑠}B$$
9	4 8	eqtrd	$$⊢\left(A\in X\wedge B\subseteq A\right)\to \left(W{\upharpoonright }_{𝑠}A\right){\upharpoonright }_{𝑠}B=W{\upharpoonright }_{𝑠}B$$