Theorem sbthlem3

Description: Lemma for sbth . (Contributed by NM, 22-Mar-1998)

	Ref	Expression
Hypotheses	sbthlem.1	$$⊢A\in \mathrm{V}$$
	sbthlem.2	$$⊢D=\left\{x|\left(x\subseteq A\wedge g\left[B\setminus f\left[x\right]\right]\subseteq A\setminus x\right)\right\}$$
Assertion	sbthlem3	$$⊢\mathrm{ran}g\subseteq A\to g\left[B\setminus f\left[\bigcup D\right]\right]=A\setminus \bigcup D$$

Proof

Step	Hyp	Ref	Expression
1		sbthlem.1	$$⊢A\in \mathrm{V}$$
2		sbthlem.2	$$⊢D=\left\{x|\left(x\subseteq A\wedge g\left[B\setminus f\left[x\right]\right]\subseteq A\setminus x\right)\right\}$$
3	1 2	sbthlem2	$$⊢\mathrm{ran}g\subseteq A\to A\setminus g\left[B\setminus f\left[\bigcup D\right]\right]\subseteq \bigcup D$$
4	1 2	sbthlem1	$$⊢\bigcup D\subseteq A\setminus g\left[B\setminus f\left[\bigcup D\right]\right]$$
5	3 4	jctil	$$⊢\mathrm{ran}g\subseteq A\to \left(\bigcup D\subseteq A\setminus g\left[B\setminus f\left[\bigcup D\right]\right]\wedge A\setminus g\left[B\setminus f\left[\bigcup D\right]\right]\subseteq \bigcup D\right)$$
6		eqss	$$⊢\bigcup D=A\setminus g\left[B\setminus f\left[\bigcup D\right]\right]\leftrightarrow \left(\bigcup D\subseteq A\setminus g\left[B\setminus f\left[\bigcup D\right]\right]\wedge A\setminus g\left[B\setminus f\left[\bigcup D\right]\right]\subseteq \bigcup D\right)$$
7	5 6	sylibr	$$⊢\mathrm{ran}g\subseteq A\to \bigcup D=A\setminus g\left[B\setminus f\left[\bigcup D\right]\right]$$
8	7	difeq2d	$$⊢\mathrm{ran}g\subseteq A\to A\setminus \bigcup D=A\setminus \left(A\setminus g\left[B\setminus f\left[\bigcup D\right]\right]\right)$$
9		imassrn	$$⊢g\left[B\setminus f\left[\bigcup D\right]\right]\subseteq \mathrm{ran}g$$
10		sstr2	$$⊢g\left[B\setminus f\left[\bigcup D\right]\right]\subseteq \mathrm{ran}g\to \left(\mathrm{ran}g\subseteq A\to g\left[B\setminus f\left[\bigcup D\right]\right]\subseteq A\right)$$
11	9 10	ax-mp	$$⊢\mathrm{ran}g\subseteq A\to g\left[B\setminus f\left[\bigcup D\right]\right]\subseteq A$$
12		dfss4	$$⊢g\left[B\setminus f\left[\bigcup D\right]\right]\subseteq A\leftrightarrow A\setminus \left(A\setminus g\left[B\setminus f\left[\bigcup D\right]\right]\right)=g\left[B\setminus f\left[\bigcup D\right]\right]$$
13	11 12	sylib	$$⊢\mathrm{ran}g\subseteq A\to A\setminus \left(A\setminus g\left[B\setminus f\left[\bigcup D\right]\right]\right)=g\left[B\setminus f\left[\bigcup D\right]\right]$$
14	8 13	eqtr2d	$$⊢\mathrm{ran}g\subseteq A\to g\left[B\setminus f\left[\bigcup D\right]\right]=A\setminus \bigcup D$$