Theorem iuneq1

Description: Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998)

	Ref	Expression
Assertion	iuneq1	$$⊢A=B\to \bigcup _{x\in A}C=\bigcup _{x\in B}C$$

Proof

Step	Hyp	Ref	Expression
1		iunss1	$$⊢A\subseteq B\to \bigcup _{x\in A}C\subseteq \bigcup _{x\in B}C$$
2		iunss1	$$⊢B\subseteq A\to \bigcup _{x\in B}C\subseteq \bigcup _{x\in A}C$$
3	1 2	anim12i	$$⊢\left(A\subseteq B\wedge B\subseteq A\right)\to \left(\bigcup _{x\in A}C\subseteq \bigcup _{x\in B}C\wedge \bigcup _{x\in B}C\subseteq \bigcup _{x\in A}C\right)$$
4		eqss	$$⊢A=B\leftrightarrow \left(A\subseteq B\wedge B\subseteq A\right)$$
5		eqss	$$⊢\bigcup _{x\in A}C=\bigcup _{x\in B}C\leftrightarrow \left(\bigcup _{x\in A}C\subseteq \bigcup _{x\in B}C\wedge \bigcup _{x\in B}C\subseteq \bigcup _{x\in A}C\right)$$
6	3 4 5	3imtr4i	$$⊢A=B\to \bigcup _{x\in A}C=\bigcup _{x\in B}C$$