Theorem rankuni2

Description: The rank of a union. Part of Theorem 15.17(iv) of Monk1 p. 112. (Contributed by NM, 30-Nov-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypothesis	ranksn.1	|- A e. _V
Assertion	rankuni2	|- ( rank ` U. A ) = U_ x e. A ( rank ` x )

Proof

Step	Hyp	Ref	Expression
1		ranksn.1	|- A e. _V
2		unir1	|- U. ( R1 " On ) = _V
3	1 2	eleqtrri	|- A e. U. ( R1 " On )
4		rankuni2b	|- ( A e. U. ( R1 " On ) -> ( rank ` U. A ) = U_ x e. A ( rank ` x ) )
5	3 4	ax-mp	|- ( rank ` U. A ) = U_ x e. A ( rank ` x )