Theorem frege87d

Description: If the images of both { A } and U are subsets of U and C follows A in the transitive closure of R and B follows C in R , then B is an element of U . Similar to Proposition 87 of Frege1879 p. 66. Compare with frege87 . (Contributed by RP, 15-Jul-2020)

	Ref	Expression
Hypotheses	frege87d.r	⊢ ( 𝜑 → 𝑅 ∈ V )
	frege87d.a	⊢ ( 𝜑 → 𝐴 ∈ V )
	frege87d.b	⊢ ( 𝜑 → 𝐵 ∈ V )
	frege87d.c	⊢ ( 𝜑 → 𝐶 ∈ V )
	frege87d.ac	⊢ ( 𝜑 → 𝐴 ( t+ ‘ 𝑅 ) 𝐶 )
	frege87d.cb	⊢ ( 𝜑 → 𝐶 𝑅 𝐵 )
	frege87d.ss	⊢ ( 𝜑 → ( 𝑅 “ { 𝐴 } ) ⊆ 𝑈 )
	frege87d.he	⊢ ( 𝜑 → ( 𝑅 “ 𝑈 ) ⊆ 𝑈 )
Assertion	frege87d	⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		frege87d.r	⊢ ( 𝜑 → 𝑅 ∈ V )
2		frege87d.a	⊢ ( 𝜑 → 𝐴 ∈ V )
3		frege87d.b	⊢ ( 𝜑 → 𝐵 ∈ V )
4		frege87d.c	⊢ ( 𝜑 → 𝐶 ∈ V )
5		frege87d.ac	⊢ ( 𝜑 → 𝐴 ( t+ ‘ 𝑅 ) 𝐶 )
6		frege87d.cb	⊢ ( 𝜑 → 𝐶 𝑅 𝐵 )
7		frege87d.ss	⊢ ( 𝜑 → ( 𝑅 “ { 𝐴 } ) ⊆ 𝑈 )
8		frege87d.he	⊢ ( 𝜑 → ( 𝑅 “ 𝑈 ) ⊆ 𝑈 )
9	1 2 3 4 5 6	frege96d	⊢ ( 𝜑 → 𝐴 ( t+ ‘ 𝑅 ) 𝐵 )
10	1 2 3 9 8 7	frege77d	⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )