Theorem frege93

Description: Necessary condition for two elements to be related by the transitive closure. Proposition 93 of Frege1879 p. 70. (Contributed by RP, 2-Jul-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege91.x	⊢ 𝑋 ∈ 𝑈
	frege91.y	⊢ 𝑌 ∈ 𝑉
	frege91.r	⊢ 𝑅 ∈ 𝑊
Assertion	frege93	⊢ ( ∀ 𝑓 ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑌 ∈ 𝑓 ) ) → 𝑋 ( t+ ‘ 𝑅 ) 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		frege91.x	⊢ 𝑋 ∈ 𝑈
2		frege91.y	⊢ 𝑌 ∈ 𝑉
3		frege91.r	⊢ 𝑅 ∈ 𝑊
4		vex	⊢ 𝑓 ∈ V
5	4	frege60c	⊢ ( ∀ 𝑓 ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑌 ∈ 𝑓 ) ) → ( [ 𝑓 / 𝑓 ] 𝑅 hereditary 𝑓 → ( [ 𝑓 / 𝑓 ] ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) → [ 𝑓 / 𝑓 ] 𝑌 ∈ 𝑓 ) ) )
6		sbcid	⊢ ( [ 𝑓 / 𝑓 ] 𝑅 hereditary 𝑓 ↔ 𝑅 hereditary 𝑓 )
7		sbcid	⊢ ( [ 𝑓 / 𝑓 ] ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) ↔ ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) )
8		sbcid	⊢ ( [ 𝑓 / 𝑓 ] 𝑌 ∈ 𝑓 ↔ 𝑌 ∈ 𝑓 )
9	7 8	imbi12i	⊢ ( ( [ 𝑓 / 𝑓 ] ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) → [ 𝑓 / 𝑓 ] 𝑌 ∈ 𝑓 ) ↔ ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) → 𝑌 ∈ 𝑓 ) )
10	5 6 9	3imtr3g	⊢ ( ∀ 𝑓 ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑌 ∈ 𝑓 ) ) → ( 𝑅 hereditary 𝑓 → ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) → 𝑌 ∈ 𝑓 ) ) )
11	10	axc4i	⊢ ( ∀ 𝑓 ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑌 ∈ 𝑓 ) ) → ∀ 𝑓 ( 𝑅 hereditary 𝑓 → ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) → 𝑌 ∈ 𝑓 ) ) )
12	1 2 3	frege90	⊢ ( ( ∀ 𝑓 ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑌 ∈ 𝑓 ) ) → ∀ 𝑓 ( 𝑅 hereditary 𝑓 → ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) → 𝑌 ∈ 𝑓 ) ) ) → ( ∀ 𝑓 ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑌 ∈ 𝑓 ) ) → 𝑋 ( t+ ‘ 𝑅 ) 𝑌 ) )
13	11 12	ax-mp	⊢ ( ∀ 𝑓 ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝑓 ) → ( 𝑅 hereditary 𝑓 → 𝑌 ∈ 𝑓 ) ) → 𝑋 ( t+ ‘ 𝑅 ) 𝑌 )