Theorem ssiun2s

Description: Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003)

	Ref	Expression
Hypothesis	ssiun2s.1	$$⊢x=C\to B=D$$
Assertion	ssiun2s	$$⊢C\in A\to D\subseteq \bigcup _{x\in A}B$$

Proof

Step	Hyp	Ref	Expression
1		ssiun2s.1	$$⊢x=C\to B=D$$
2		nfcv	$$⊢\underset{\_}{Ⅎ}xC$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}xD$$
4		nfiu1	$$⊢\underset{\_}{Ⅎ}x\bigcup _{x\in A}B$$
5	3 4	nfss	$$⊢ℲxD\subseteq \bigcup _{x\in A}B$$
6	1	sseq1d	$$⊢x=C\to \left(B\subseteq \bigcup _{x\in A}B\leftrightarrow D\subseteq \bigcup _{x\in A}B\right)$$
7		ssiun2	$$⊢x\in A\to B\subseteq \bigcup _{x\in A}B$$
8	2 5 6 7	vtoclgaf	$$⊢C\in A\to D\subseteq \bigcup _{x\in A}B$$