Theorem ssiun2sf

Description: Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016)

	Ref	Expression
Hypotheses	ssiun2sf.1	$$⊢\underset{\_}{Ⅎ}xA$$
	ssiun2sf.2	$$⊢\underset{\_}{Ⅎ}xC$$
	ssiun2sf.3	$$⊢\underset{\_}{Ⅎ}xD$$
	ssiun2sf.4	$$⊢x=C\to B=D$$
Assertion	ssiun2sf	$$⊢C\in A\to D\subseteq \bigcup _{x\in A}B$$

Proof

Step	Hyp	Ref	Expression
1		ssiun2sf.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		ssiun2sf.2	$$⊢\underset{\_}{Ⅎ}xC$$
3		ssiun2sf.3	$$⊢\underset{\_}{Ⅎ}xD$$
4		ssiun2sf.4	$$⊢x=C\to B=D$$
5	2 1	nfel	$$⊢ℲxC\in A$$
6		nfiu1	$$⊢\underset{\_}{Ⅎ}x\bigcup _{x\in A}B$$
7	3 6	nfss	$$⊢ℲxD\subseteq \bigcup _{x\in A}B$$
8	5 7	nfim	$$⊢Ⅎx\left(C\in A\to D\subseteq \bigcup _{x\in A}B\right)$$
9		eleq1	$$⊢x=C\to \left(x\in A\leftrightarrow C\in A\right)$$
10	4	sseq1d	$$⊢x=C\to \left(B\subseteq \bigcup _{x\in A}B\leftrightarrow D\subseteq \bigcup _{x\in A}B\right)$$
11	9 10	imbi12d	$$⊢x=C\to \left(\left(x\in A\to B\subseteq \bigcup _{x\in A}B\right)\leftrightarrow \left(C\in A\to D\subseteq \bigcup _{x\in A}B\right)\right)$$
12		ssiun2	$$⊢x\in A\to B\subseteq \bigcup _{x\in A}B$$
13	2 8 11 12	vtoclgf	$$⊢C\in A\to \left(C\in A\to D\subseteq \bigcup _{x\in A}B\right)$$
14	13	pm2.43i	$$⊢C\in A\to D\subseteq \bigcup _{x\in A}B$$