Theorem frege109

Description: The property of belonging to the R -sequence beginning with X is hereditary in the R -sequence. Proposition 109 of Frege1879 p. 74. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege109.x	⊢ 𝑋 ∈ 𝑈
	frege109.r	⊢ 𝑅 ∈ 𝑉
Assertion	frege109	⊢ 𝑅 hereditary ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑋 } )

Proof

Step	Hyp	Ref	Expression
1		frege109.x	⊢ 𝑋 ∈ 𝑈
2		frege109.r	⊢ 𝑅 ∈ 𝑉
3		frege75	⊢ ( ∀ 𝑦 ( 𝑦 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑋 } ) → ∀ 𝑧 ( 𝑦 𝑅 𝑧 → 𝑧 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑋 } ) ) ) → 𝑅 hereditary ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑋 } ) )
4		vex	⊢ 𝑦 ∈ V
5		vex	⊢ 𝑧 ∈ V
6	1 4 5 2	frege108	⊢ ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑦 → ( 𝑦 𝑅 𝑧 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑧 ) )
7		df-br	⊢ ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑦 ↔ ⟨ 𝑋 , 𝑦 ⟩ ∈ ( ( t+ ‘ 𝑅 ) ∪ I ) )
8	1	elexi	⊢ 𝑋 ∈ V
9	8 4	elimasn	⊢ ( 𝑦 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑋 } ) ↔ ⟨ 𝑋 , 𝑦 ⟩ ∈ ( ( t+ ‘ 𝑅 ) ∪ I ) )
10	7 9	bitr4i	⊢ ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑦 ↔ 𝑦 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑋 } ) )
11		df-br	⊢ ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑧 ↔ ⟨ 𝑋 , 𝑧 ⟩ ∈ ( ( t+ ‘ 𝑅 ) ∪ I ) )
12	8 5	elimasn	⊢ ( 𝑧 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑋 } ) ↔ ⟨ 𝑋 , 𝑧 ⟩ ∈ ( ( t+ ‘ 𝑅 ) ∪ I ) )
13	11 12	bitr4i	⊢ ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑧 ↔ 𝑧 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑋 } ) )
14	13	imbi2i	⊢ ( ( 𝑦 𝑅 𝑧 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑧 ) ↔ ( 𝑦 𝑅 𝑧 → 𝑧 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑋 } ) ) )
15	6 10 14	3imtr3i	⊢ ( 𝑦 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑋 } ) → ( 𝑦 𝑅 𝑧 → 𝑧 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑋 } ) ) )
16	15	alrimiv	⊢ ( 𝑦 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑋 } ) → ∀ 𝑧 ( 𝑦 𝑅 𝑧 → 𝑧 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑋 } ) ) )
17	3 16	mpg	⊢ 𝑅 hereditary ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑋 } )