Theorem frege94

Description: Looking one past a pair related by transitive closure of a relation. Proposition 94 of Frege1879 p. 70. (Contributed by RP, 2-Jul-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege94.x	⊢ 𝑋 ∈ 𝑈
	frege94.z	⊢ 𝑍 ∈ 𝑉
	frege94.r	⊢ 𝑅 ∈ 𝑊
Assertion	frege94	⊢ ( ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ∀ 𝑓 ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑍 ∈ 𝑓 ) ) ) ) → ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege94.x	⊢ 𝑋 ∈ 𝑈
2		frege94.z	⊢ 𝑍 ∈ 𝑉
3		frege94.r	⊢ 𝑅 ∈ 𝑊
4	1 2 3	frege93	⊢ ( ∀ 𝑓 ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑍 ∈ 𝑓 ) ) → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 )
5		frege7	⊢ ( ( ∀ 𝑓 ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑍 ∈ 𝑓 ) ) → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) → ( ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ∀ 𝑓 ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑍 ∈ 𝑓 ) ) ) ) → ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) ) ) )
6	4 5	ax-mp	⊢ ( ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ∀ 𝑓 ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑍 ∈ 𝑓 ) ) ) ) → ( 𝑌 𝑅 𝑍 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑋 ( t+ ‘ 𝑅 ) 𝑍 ) ) )