Metamath Proof Explorer

Theorem iuneq1

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

Ref Expression
Assertion iuneq1 
|- ( A = B -> U_ x e. A C = U_ x e. B C )

Proof

Step Hyp Ref Expression
1 iunss1 
 |-  ( A C_ B -> U_ x e. A C C_ U_ x e. B C )
2 iunss1 
 |-  ( B C_ A -> U_ x e. B C C_ U_ x e. A C )
3 1 2 anim12i 
 |-  ( ( A C_ B /\ B C_ A ) -> ( U_ x e. A C C_ U_ x e. B C /\ U_ x e. B C C_ U_ x e. A C ) )
4 eqss 
 |-  ( A = B <-> ( A C_ B /\ B C_ A ) )
5 eqss 
 |-  ( U_ x e. A C = U_ x e. B C <-> ( U_ x e. A C C_ U_ x e. B C /\ U_ x e. B C C_ U_ x e. A C ) )
6 3 4 5 3imtr4i 
 |-  ( A = B -> U_ x e. A C = U_ x e. B C )