Theorem frege129

Description: If the procedure R is single-valued and Y belongs to the R -sequence begining 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 begining 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 e. U
	frege123.y	|- Y e. V
	frege124.m	|- M e. W
	frege124.r	|- R e. S
Assertion	frege129	|- ( Fun `' `' R -> ( ( -. Y ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) Y ) -> ( Y R X -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege123.x	|- X e. U
2		frege123.y	|- Y e. V
3		frege124.m	|- M e. W
4		frege124.r	|- R e. S
5	3 2 1 4	frege111	|- ( M ( ( t+ ` R ) u. _I ) Y -> ( Y R X -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) )
6	1 2 3 4	frege128	|- ( ( M ( ( t+ ` R ) u. _I ) Y -> ( Y R X -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) -> ( Fun `' `' R -> ( ( -. Y ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) Y ) -> ( Y R X -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) ) )
7	5 6	ax-mp	|- ( Fun `' `' R -> ( ( -. Y ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) Y ) -> ( Y R X -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) )