Theorem frege83d

Description: If the image of the union of U and V is a subset of the union of U and V , A is an element of U and B follows A in the transitive closure of R , then B is an element of the union of U and V . Similar to Proposition 83 of Frege1879 p. 65. Compare with frege83 . (Contributed by RP, 15-Jul-2020)

	Ref	Expression
Hypotheses	frege83d.r	|- ( ph -> R e. _V )
	frege83d.a	|- ( ph -> A e. U )
	frege83d.b	|- ( ph -> B e. _V )
	frege83d.ab	|- ( ph -> A ( t+ ` R ) B )
	frege83d.he	|- ( ph -> ( R " ( U u. V ) ) C_ ( U u. V ) )
Assertion	frege83d	|- ( ph -> B e. ( U u. V ) )

Proof

Step	Hyp	Ref	Expression
1		frege83d.r	|- ( ph -> R e. _V )
2		frege83d.a	|- ( ph -> A e. U )
3		frege83d.b	|- ( ph -> B e. _V )
4		frege83d.ab	|- ( ph -> A ( t+ ` R ) B )
5		frege83d.he	|- ( ph -> ( R " ( U u. V ) ) C_ ( U u. V ) )
6		ssun1	|- U C_ ( U u. V )
7	6 2	sselid	|- ( ph -> A e. ( U u. V ) )
8	1 7 3 4 5	frege81d	|- ( ph -> B e. ( U u. V ) )