Theorem iuneq1i

Description: Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021) Remove DV conditions. (Revised by GG, 1-Sep-2025)

	Ref	Expression
Hypothesis	iuneq1i.1	|- A = B
Assertion	iuneq1i	|- U_ x e. A C = U_ x e. B C

Proof

Step	Hyp	Ref	Expression
1		iuneq1i.1	|- A = B
2	1	eleq2i	|- ( x e. A <-> x e. B )
3	2	anbi1i	|- ( ( x e. A /\ t e. C ) <-> ( x e. B /\ t e. C ) )
4	3	rexbii2	|- ( E. x e. A t e. C <-> E. x e. B t e. C )
5	4	abbii	|- { t | E. x e. A t e. C } = { t | E. x e. B t e. C }
6		df-iun	|- U_ x e. A C = { t | E. x e. A t e. C }
7		df-iun	|- U_ x e. B C = { t | E. x e. B t e. C }
8	5 6 7	3eqtr4i	|- U_ x e. A C = U_ x e. B C