Theorem cofss

Description: Cofinality for a subset. (Contributed by Scott Fenton, 13-Mar-2025)

	Ref	Expression
Hypotheses	cofss.1	$$⊢\phi \to A\subseteq \mathrm{No}$$
	cofss.2	$$⊢\phi \to B\subseteq A$$
Assertion	cofss	$$⊢\phi \to \forall x\in B\exists y\in Ax{\le }_{s}y$$

Proof

Step	Hyp	Ref	Expression
1		cofss.1	$$⊢\phi \to A\subseteq \mathrm{No}$$
2		cofss.2	$$⊢\phi \to B\subseteq A$$
3	2	sselda	$$⊢\left(\phi \wedge z\in B\right)\to z\in A$$
4	2 1	sstrd	$$⊢\phi \to B\subseteq \mathrm{No}$$
5	4	sselda	$$⊢\left(\phi \wedge z\in B\right)\to z\in \mathrm{No}$$
6		slerflex	$$⊢z\in \mathrm{No}\to z{\le }_{s}z$$
7	5 6	syl	$$⊢\left(\phi \wedge z\in B\right)\to z{\le }_{s}z$$
8		breq2	$$⊢y=z\to \left(z{\le }_{s}y\leftrightarrow z{\le }_{s}z\right)$$
9	8	rspcev	$$⊢\left(z\in A\wedge z{\le }_{s}z\right)\to \exists y\in Az{\le }_{s}y$$
10	3 7 9	syl2anc	$$⊢\left(\phi \wedge z\in B\right)\to \exists y\in Az{\le }_{s}y$$
11	10	ralrimiva	$$⊢\phi \to \forall z\in B\exists y\in Az{\le }_{s}y$$
12		breq1	$$⊢x=z\to \left(x{\le }_{s}y\leftrightarrow z{\le }_{s}y\right)$$
13	12	rexbidv	$$⊢x=z\to \left(\exists y\in Ax{\le }_{s}y\leftrightarrow \exists y\in Az{\le }_{s}y\right)$$
14	13	cbvralvw	$$⊢\forall x\in B\exists y\in Ax{\le }_{s}y\leftrightarrow \forall z\in B\exists y\in Az{\le }_{s}y$$
15	11 14	sylibr	$$⊢\phi \to \forall x\in B\exists y\in Ax{\le }_{s}y$$