Theorem frege81d

Description: If the image of U is a subset U , A is an element of U and B follows A in the transitive closure of R , then B is an element of U . Similar to Proposition 81 of Frege1879 p. 63. Compare with frege81 . (Contributed by RP, 15-Jul-2020)

	Ref	Expression
Hypotheses	frege81d.r	|- ( ph -> R e. _V )
	frege81d.a	|- ( ph -> A e. U )
	frege81d.b	|- ( ph -> B e. _V )
	frege81d.ab	|- ( ph -> A ( t+ ` R ) B )
	frege81d.he	|- ( ph -> ( R " U ) C_ U )
Assertion	frege81d	|- ( ph -> B e. U )

Proof

Step	Hyp	Ref	Expression
1		frege81d.r	|- ( ph -> R e. _V )
2		frege81d.a	|- ( ph -> A e. U )
3		frege81d.b	|- ( ph -> B e. _V )
4		frege81d.ab	|- ( ph -> A ( t+ ` R ) B )
5		frege81d.he	|- ( ph -> ( R " U ) C_ U )
6	2	elexd	|- ( ph -> A e. _V )
7	2	snssd	|- ( ph -> { A } C_ U )
8		imass2	|- ( { A } C_ U -> ( R " { A } ) C_ ( R " U ) )
9	7 8	syl	|- ( ph -> ( R " { A } ) C_ ( R " U ) )
10	9 5	sstrd	|- ( ph -> ( R " { A } ) C_ U )
11	1 6 3 4 5 10	frege77d	|- ( ph -> B e. U )