Theorem frege122

Description: If X is a result of an application of the single-valued procedure R to Y , then every result of an application of the procedure R to Y belongs to the R -sequence beginning with X . Proposition 122 of Frege1879 p. 79. (Contributed by RP, 8-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege116.x	⊢ 𝑋 ∈ 𝑈
	frege118.y	⊢ 𝑌 ∈ 𝑉
	frege120.a	⊢ 𝐴 ∈ 𝑊
Assertion	frege122	⊢ ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ( 𝑌 𝑅 𝐴 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege116.x	⊢ 𝑋 ∈ 𝑈
2		frege118.y	⊢ 𝑌 ∈ 𝑉
3		frege120.a	⊢ 𝐴 ∈ 𝑊
4	3	frege112	⊢ ( 𝐴 = 𝑋 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝐴 )
5	1 2 3	frege121	⊢ ( ( 𝐴 = 𝑋 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝐴 ) → ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ( 𝑌 𝑅 𝐴 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝐴 ) ) ) )
6	4 5	ax-mp	⊢ ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ( 𝑌 𝑅 𝐴 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝐴 ) ) )