Theorem frege129

Description: If the procedure R is single-valued and Y belongs to the R -sequence beginning with M or precedes M in the R -sequence, then every result of an application of the procedure R to Y belongs to the R -sequence beginning with M or precedes M in the R -sequence. Proposition 129 of Frege1879 p. 83. (Contributed by RP, 9-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege123.x	⊢ 𝑋 ∈ 𝑈
	frege123.y	⊢ 𝑌 ∈ 𝑉
	frege124.m	⊢ 𝑀 ∈ 𝑊
	frege124.r	⊢ 𝑅 ∈ 𝑆
Assertion	frege129	⊢ ( Fun ◡ ◡ 𝑅 → ( ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) → ( 𝑌 𝑅 𝑋 → ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege123.x	⊢ 𝑋 ∈ 𝑈
2		frege123.y	⊢ 𝑌 ∈ 𝑉
3		frege124.m	⊢ 𝑀 ∈ 𝑊
4		frege124.r	⊢ 𝑅 ∈ 𝑆
5	3 2 1 4	frege111	⊢ ( 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑋 → ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 ) ) )
6	1 2 3 4	frege128	⊢ ( ( 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 → ( 𝑌 𝑅 𝑋 → ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 ) ) ) → ( Fun ◡ ◡ 𝑅 → ( ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) → ( 𝑌 𝑅 𝑋 → ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 ) ) ) ) )
7	5 6	ax-mp	⊢ ( Fun ◡ ◡ 𝑅 → ( ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) → ( 𝑌 𝑅 𝑋 → ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 ) ) ) )